Brownian Polymers in Poissonian Environment: a survey
aa r X i v : . [ m a t h . P R ] M a y Brownian Polymers in Poissonian Environment:a survey.
Francis Comets, Cl´ement Cosco
Universit´e Paris DiderotLaboratoire de Probabilit´es, Statistique et Mod´elisationLPSM (UMR 8001 CNRS, SU, UPD)Bˆatiment Sophie Germain, 8 place Aur´elie Nemours, 75013 Paris [email protected], [email protected]
May 29, 2018 bstract
We consider a space-time continuous directed polymer in random environment. The path isBrownian and the medium is Poissonian. We review many results obtained in the last decade,and also we present new ones. In this fundamental setup, we can make use of fine formulas andstrong tools from stochastic analysis for Gaussian or Poisson measure, together with martingaletechniques. These notes cover the matter of a course presented during the Jean-Morlet chair2017 of CIRM ”Random Structures in Statistical Mechanics and Mathematical Physics” inMarseille.
Keywords:
Directed polymers, random environment; weak disorder, intermediate disorder,strong disorder; free energy; Poisson processes, martingales.
AMS 2010 subject classifications:
Primary 60K37. Secondary 60Hxx, 82A51, 82D30. ontents L -region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.4 Relations between the different critical temperatures . . . . . . . . . . . . . . . 18 Z t . . . . . . . . . . . . . . . . . . . . . . 255.3 The replica overlap and quenched overlaps . . . . . . . . . . . . . . . . . . . . . 255.4 Endpoint localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.5 Favorite path and path localization . . . . . . . . . . . . . . . . . . . . . . . . . 29 ( β, ν ) -plane, d ≥ PPE
10 The Intermediate Regime ( d = 1 ) 50 Introduction
This survey is based on a course presented by the first author at the Research School inMarseille, March 6-10, 2017. The school was organized by the chair holders, Kostya Khaninand Senya Shlosman, of the Jean-Morlet chair 2017 of CIRM,Random Structures in Statistical Mechanics and Mathematical Physics.The model is a space-time continuous directed polymer in random environment. In this regard,it is one of the most basic such model and it plays a fundamental role. Directed polymers aredescribed by random paths, which are influenced by randomly located impurities which maybe attractive or repellent. Such models have been widely considered in statistical physics,disordered systems and stochastic processes.As an informal definition we model the polymer by a random path x = ( x ( t ); t ≥
0) takingvalues in R d and interacting with time-space Poisson points ( t i , x i ) called environment. Thepath sees such a point if at time t i it is located within a fixed distance r from x i . Denoting by t ( x ) = P i : t i ≤ t | x ( t i ) − x i |≤ r the number of Poisson points seen by the path x up to time t , themodel with time horizon t at inverse-temperature parameter β is associated to the Hamiltonian − Z t | ˙ x ( s ) | ds + β t ( x ) . In this model where the path is Brownian and the medium is Poissonian, we benefit fromnice formulas and strong tools from stochastic calculus for Gaussian or Poisson measure andmartingale techniques.The notes are essentially based on references [19, 18, 20, 22], gathering and unifying thematter scattered in these references, and containing novel contributions and perspectives asemphasized below. It also parallels the book [17] which deals similar models in the discreteframework, and we warn the reader of the existence of many results available for one particularmodel but not for the others. We do not reproduce all details or computations, but we rathertry to give the general picture and the essential arguments.Let us mention the main highlights in this survey and also the new results:1. We establish in section 3 a fine continuity estimate under spatial shifts for the limit ofthe martingale. This is achieved by a smart use of mirror coupling.2. Section 4 contains a nice original account on directional free energy. We develop a fullapproach of disorder strength based on directional free energy.3. In section 7 we develop an original approach to diffusivity at weak disorder, based onCamer´on-Martin transformation (see theorem 7.2.2).4. Section 10 is dedicated to the intermediate disorder regime and KPZ equation. We givea synthetic account with all the central ideas.The detailed matter and the organization appear most clearly in the table of contents,which is a useful source to follow the line all through the notes.3
Free energy and phase transition
Notations and conventions: all through the notes, we will use the same symbols P, P , . . . todenote probability measures and mathematical expectations; e.g., P [ X ] is the P -expectationof the random variable X .In this section, we introduce the model and two central thermodynamic quantities, thequenched and the annealed free energies. The model is defined as a Brownian motion in a random potential. • The free measure : ( B = { B t } t ≥ , P x ) is a Brownian motion on the d -dimensional Eu-clidean space R d starting from x ∈ R d . We will use short notation P = P . • The random environment: ω = P i δ ( T i ,X i ) is a Poisson point process on R + × R d withintensity measure νdtdx , where ν is a positive parameter. We suppose that ω is defined on someprobability space (Ω , G , P ), and we define G t to be the σ -field generated by the environment upto time t : ω t = ω | (0 ,t ] × R d , G t = σ (cid:16) ω t ( A ); A ∈ B ( R + × R d ) (cid:17) , (2.1.1)where B ( R + × R d ) denotes the Borel sets of R + × R d .From these two basic ingredients, we define the object we consider in the notes. Fix r > U ( x ) denote Euclidean (closed) ball in R d with radius γ − /dd r , U ( x ) = B( x, γ − /dd r ) . with γ d the volume of the unit ball, so U ( x ) has volume r d . The tube around path B is thefollowing subset of (0 , t ] × R d : V t ( B ) = { ( s, x ) : s ∈ (0 , t ] , x ∈ U ( B s ) } . (2.1.2)When the indicator function χ s,x = { x ∈ U ( B s ) } = {| x − B s | ≤ γ − /dd r } (2.1.3)has value 1 [resp., 0], the path B does see [resp. does not see] the point ( s, x ). For a fixed path B , the quantity defined by ω ( V t ) = Z (0 ,t ] × R d χ s,x ω ( ds, dx ) , (2.1.4)is the number of Poisson points seen by the path B up to time t , playing the role of t in theIntroduction. Note that under P , the variable ω ( V t ) is Poisson distributed with mean νtr d . • The polymer measure:
Fixing a realization ω of the Poisson point process and a value ofthe time horizon t >
0, we define the probability measure P β,ωt on the path space C ( R + ; R d )equipped with its Borel field by dP β,ωt = 1 Z t ( ω, β, r ) exp { βω ( V t ) } dP, (2.1.5)4 PPE β ∈ R is a parameter (the inverse temperature), where Z t = Z t ( ω, β, r ) = P [exp ( βω ( V t ))] (2.1.6)is the normalizing constant making P β,ωt a probability measure on the path space.The model has been introduced by Nobuo Yoshida as a polymer model, and first appearedin [19] in the literature. For β > β the model relates to Brownian motion in Poissonian obstacles [25, 61] which can be tracedback to works of Smoluchowski [14]. Here we consider a directed version, in contrast to crossings[67, 66, 68, 69] where the path is stretched ballistically. Our model with β → + ∞ is related toEuclidean first passage percolation [29, 28] with exponent α = 2 therein.Also, for a branching Brownian motion in random medium [57, 58], Z t is equal to the meanpopulation size in the medium given by ω . We first recall three basic formulas that we will use repeatedly. • For all non-negative and all non-positive measurable functions h on R + × R d , the Poissonformula for exponential moments (chapter 3. of [41]) writes P h e R h ( s,x ) ω t ( dsdx ) i = exp Z ]0 ,t ] × R d νdsdx (cid:16) e h ( s,x ) − (cid:17) . (2.2.7)The formula remains true when h is replaced by ih , for any real integrable function h . • Introducing the notation λ ( β ) = e β − , (2.2.8)the linearization formula for Bernoulli writes e β A − e β − A = λ ( β ) A . (2.2.9) • For all s ≥
0, we have Z R d χ s,x d x = r d . (2.2.10) It is defined as the rate of growth of the partition function, and it is a self-averaging property.
Theorem 2.3.1.
The quenched free energy p ( β, ν ) = lim t →∞ t ln Z t ( ω, β, r ) exists a.s. and in L p -norm for all p ≥ , and is deterministic, p ( β, ν ) = sup t> t P [ln Z t ] . Remark 2.3.2.
We omit the parameter r > from the notation for the free energy. Thereason is that, in contrast to β and ν , it is kept fixed most of the time. PPE ✷ Let θ t,x the space-time shift operator on the environment space, θ t,x (cid:0) X i δ ( T i ,X i ) (cid:1) = X i δ ( T i − t,X i − x ) . By Markov property of the Brownian motion, we have for s, t ≥ Z t + s = P h e βω ( V t ) e βω ( V t + s \ V t ) i = P h e βω ( V t ) P h e βω ( V t + s \ V t ) (cid:12)(cid:12) B t ii = P h e βω ( V t ) Z s ◦ θ t,B t i (2.3.11)= Z t × P β,ωt [ Z s ◦ θ t,B t ] , (2.3.12)a remarkable identity expressing the Markov structure of the model. Let u ( t ) = P [ln Z t ]. Bythe independence property of Poisson points, ω | ] s,t ] is independent of G s for all 0 ≤ s ≤ t . Then,denoting by P G t the conditional expectation and conditional probability given G t , we have u ( t + s ) = P (cid:2) ln P β,ωt [ Z s ◦ θ t,B t ) ] (cid:3) + P ln Z t Jensen ≥ P P β,ωt [ln Z s ◦ θ t,B t ) ] + u ( t )= PP G t P β,ωt [ln Z s ◦ θ t,B t ) ] + u ( t ) Fubini = P (cid:2) P β,ωt [ P G t [ln Z s ◦ θ t,B t ) ]] (cid:3) + u ( t )= P (cid:2) P β,ωt [ u ( s )] (cid:3) + u ( t ) ( ω shift invariant)= u ( s ) + u ( t ) . Hence the function u ( t ) is superadditive. By the superadditive lemma, we get the existence ofthe limit lim t →∞ u ( t ) t = sup t> u ( t ) t . Now, anticipating the concentration inequality (6.3.11) and the continuous time bridging(6.3.14), we derive that 1 t (cid:0) ln Z t − P [ln Z t ] (cid:1) −→ L p for all p ≥ We compute the expectation of the partition function over the medium using (2.1.4) and Fubini, P [ Z t ] = P P h e β R χ s,x ω t ( ds,dx ) i (2.2.7) = P exp Z ]0 ,t ] × R d (cid:16) e β R χ s,x − (cid:17) νdsdx (2.2.9) = P " exp λ ( β ) Z ]0 ,t ] × R d χ s,x νdsdx = exp { tνλr d } . (2.4.13) PPE P [ Z t ] grows in time at exponential rate p (1) ( β, ν ) = νλ ( β ) r d . More generally, it is naturalto consider the rate of growth of the s -th moment of the partition function, p ( s ) ( β, ν ) = lim t →∞ st ln P [ Z st ] , s > . By H¨older inequality, k Z k r ≤ k Z k s for r ≤ s , these rates are non-decreasing in s , and for integervalues, they can be expressed by handy variational formulas using large deviation theory. ByJensen’s inequality, we have for all t t P [ln Z t ] ≤ t ln P [ Z t ] = p (1) ( ν, β ) , (2.4.14)yielding the so-called annealed bound : p ( β, ν ) ≤ p (1) ( ν, β ) . Summarizing the above, we have a chain of inequalities p ( β, ν ) ≤ p (1) ( β, ν ) ≤ . . . ≤ p ( k ) ( β, ν ) ≤ p ( k +1) ( β, ν ) ≤ . . . . It is commun folklore that in a large class of models, the first inequalities in the above chainare equalities, while they become strict from k ∗ = inf { k ≥ p ( k ) ( β, ν ) < p ( k +1) ( β, ν ) (withthe convention p (0) = p ). Considering the sequence of rates ( p ( k ) ; k ≥
1) is classical approachto intermittency [13, 37, 45] and sect. 2.4 of [11].In the directed case, we focus at k = 0 , Proposition 2.4.1.
Basic properties of the free energy:1. For β = 0 , ν > , we have βνr d < p ( β, ν ) ≤ νλ ( β ) r d .2. β → p ( β, ν ) is convex.3. The excess free energy ψ ( β, ν ) = νλ ( β ) r d − p ( β, ν ) (2.4.15) is non-decreasing in | β | and in ν . It is jointly continuous. ✷ The second inequality in item 1 is the annealed bound. The first one follows from aninifinite-dimensional version of Jensen’s inequality; this version being curiously overlooked inthe literature, we recall the full statement:
Lemma 2.4.2 (Lemma A.1 in [50]) . Let g be a bounded measurable function on a product space X × Y , µ a probability measure on X and ρ a probability measure on Y . Then ln Z X e R Y g ( x,y ) dρ ( y ) dµ ( x ) ≤ Z Y (cid:20) ln Z X e g ( x,y ) dµ ( x ) (cid:21) dρ ( y ) . We apply it with ρ = P , µ = P, g ( x, y ) = βω ( V t ) to get the desired bound . However thisbound is not so great here, since the simple one p ( β, ν ) ≥ t − P [ln Z t ] for a fixed t (which comesfrom superadditivity of u ( t )) is not linear, but strictly convex in β and then already better.Item 2 is the standard convexity of free energy, ∂ ∂β ln Z t = Var P β,ωt (cid:0) ω ( V t )) (cid:1) > , We explain in this note why the Lemma is an infinite-dimensional version of Jensen’s inequality: the func-tional ψ ( f ) = ln R X e f ( x ) dµ ( x ) is convex, and the function f ( · ) = g ( · , y ) is randomly chosen with ρ ( dy ). PPE P β,ωt denotes the variance under the polymer measure in a fixed environment ω .We now turn towards item 3, in the case β ≥ ν, ∆ >
0, we note that the superposition ω + ˆ ω of two independent PPP with intensities ν and ∆ is a PPP with intensity ν + ∆. Writing P the expectation over both variables ω, ˆ ω , we compute by conditioning P ln Z t ( ω ) β ≥ ≤ P ln Z t ( ω + ˆ ω )= PP [ln Z t ( ω + ˆ ω ) | ω ] Jensen ≤ P ln P [ Z t ( ω + ˆ ω ) | ω ]= P ln Z t ( ω ) + t ∆ λ ( β ) r d . This proves monotonicity of ψ in ν . This proves at the same time continuity in ν (locallyuniformly in β ) and the joint continuity in ( β, ν ).The plain identity P [ Y f ( Y )] = θ P [ f ( Y + 1)] for a r.v. Y distributed as a Poisson law withmean θ has a counterpart for PPP, an integration by parts formula known as Slivnyak-Meckeformula (e.g., p.50 in [60] or th. 4.1 in [41]): Let M be the space of point measures on (0 , t ] × R d and h : R + × R d × M → R + measurable, then P (cid:20)Z h ( s, x ; ω t ) ω t ( ds, dx ) (cid:21) = Z (0 ,t ] × R d P (cid:2) h ( s, x ; ω t + δ s,x ) (cid:3) νdsdx . (2.4.16)With this in hand, we can show monotonicity of ψ in β : ∂∂β P (cid:2) ln Z t (cid:3) = P P β,ωt [ ω ( V t )]= P Z ω t ( dsdx ) P (cid:2) χ s,x e βω ( V t ) (cid:3) Z t (2.4.16) = P Z (0 ,t ] × R d νdsdx P (cid:2) χ s,x e β ( ω ( V t )+ δ s,x ) (cid:3) P (cid:2) e β ( ω ( V t )+ δ s,s ) (cid:3) (2.2.9) = P Z (0 ,t ] × R d νdsdx P (cid:2) e β χ s,x e βω ( V t ) (cid:3) P (cid:2)(cid:0) λ ( β ) χ s,x + 1 (cid:1) e βω ( V t ) (cid:3) = P Z (0 ,t ] × R d νdsdx e β P β,ωt [ χ s,x ]1 + λ ( β ) P β,ωt [ χ s,x ] . (2.4.17)Define ψ t ( β, ν ) = t − P h νλr d − ln Z t i . (2.4.18)With the identity ∂∂β νλ ( β ) r d t = νe β r d t = e β Z (0 ,t ] × R d νdsdxP β,ωt [ χ s,x ]we obtain ∂∂β ψ t ( β, ν ) = 1 t e β λ ( β ) ν Z (0 ,t ] × R d dsdx P P β,ωt [ χ s,x ] λ ( β ) P β,ωt [ χ s,x ] , (2.4.19)which has the sign of β . So the limit ψ of ψ t is increasing in | β | . PPE An important consequence of monotonicity and continuity of ψ in | β | in Proposition 2.4.1 is theexistence and uniqueness of the critical temperatures introduced in the next statement, whichis a direct consequence of the above. Theorem 2.5.1.
There exist β + c ( ν ) , β − c ( ν ) with −∞ ≤ β − c ≤ ≤ β + c ≤ + ∞ such that (cid:26) ψ ( β, ν ) = 0 if β ∈ [ β − c , β + c ] ψ ( β, ν ) > β < β − c or β > β + c . (2.5.20) Moreover, | β ± c ( ν ) | is non-increasing in ν . These values β + c ( ν ) , β − c ( ν ) are called critical (inverse) temperatures at density ν (theydepend on r as well). The domains in the ( β, ν )-half-plane defined by the first and second linein (2.5.20) are called high and low temperature region respectively. The boundary between thetwo regions is called the critical line , and a phase transition in the statistical mechanics senseoccurs: the quenched free energy p ( β, ν ) is equal to the annealed free energy p (1) ( β, ν ) = νλr d – an analytic function – but analyticity of p ( β, ν ) breaks down when crossing the critical line.To summarize our finding, we define the high temperature region and the low temperatureregion D = { ( β, ν ) : ψ ( β, ν ) = 0 } , L = { ( β, ν ) : ψ ( β, ν ) > } . They are are delimited by the critical lines β − c ( ν ) and β + c ( ν ) from Definition 2.5.1. In the nextsections we will discuss non-triviality of the critical lines, as well as fine properties. In section5 we will understand that they correspond to delocalized or localized behavior respectively. Weak Disorder, Strong Disorder
In this section, we introduce a natural martingale that will play an important role in manyresults concerning the asymptotic behavior of the polymer.For any fixed path of the brownian motion, { ω ( V t ) } t ≥ is a Poisson process of intensity νr d and has associated exponential martingales { exp (cid:0) βω ( V t ) − λ ( β ) νr d t (cid:1) } t ≥ . Hence, for t ≥ normalized partition function W t = e − λ ( β ) νr d t Z t , (3.1.1)defines a positive, mean 1, c`adl`ag martingale with respect to {G t } t ≥ .By Doob’s martingale convergence theorem [51, Chapter 2, Corollary 2.11], we get theexistence of a random variable W ∞ such that W ∞ = lim t →∞ W t a.s. (3.1.2) Theorem 3.1.1.
There is a dichotomy: either the limit W ∞ is almost-surely positive, or it isalmost-surely zero. Otherwise stated, we have either P { W ∞ > } = 1 , (3.1.3) or P { W ∞ = 0 } = 1 . (3.1.4) Proof.
Denote by e t the renormalized weight e t = exp( βω ( V t ( B )) − λ ( β ) νr d t ) (3.1.5)By the Markovian property (2.3.11), we get that for all positive times t and s , W s + t = P [ e t W s ◦ θ t,B t ] . (3.1.6)In Section 3.2.1, we will justify that one can take the limit as s → ∞ in this equality, in orderto get that W ∞ = P [ e t W ∞ ◦ θ t,B t ] . (3.1.7)Then, notice that (3.1.7) also writes W ∞ = W t Z R d P β,ωt ( B t ∈ d x ) W ∞ ◦ θ t,x . (3.1.8)Since W t > P -a.s and since P β,ωt has positive density with respect to Lebesgue’s measure, weobtain by (3.1.8) that ∀ t > , { W ∞ = 0 } = { W ∞ ◦ θ t,x = 0 , x -a.e. } , or, equivalently, { W ∞ = 0 } = (cid:26)Z R d P ( B t ∈ d x ) W ∞ ◦ θ t,x = 0 (cid:27) . PPE σ -field G [ t, ∞ ) = σ (cid:0) ω ( A ); A ∈ B ([ t, ∞ ) × R d ) (cid:1) completed by null sets, so { W ∞ = 0 } ∈ ∞ \ t> G [ t, ∞ ) . The theorem now follows from Komogorov’s 0-1 law.This dichotomy calls for a definition.
Definition 3.1.2.
We say that the polymer is in the weak disorder phase when W ∞ > almost surely. We say it is in the strong disorder phase when W ∞ = 0 almost surely. The phase diagram is connected in the β -parameter space. Theorem 3.1.3.
There exist two critical parameters ¯ β − c ∈ [ −∞ , and ¯ β + c ∈ [0 , ∞ ] , dependingonly on ν, r and d , such that • For all β ∈ ( ¯ β − c , ¯ β + c ) ∪ { } , the polymer belongs to the weak disorder phase. • For all β ∈ R \ [ ¯ β − c , ¯ β + c ] , the polymer belongs to the strong disorder phase.Proof. Let θ be a real number in (0 ,
1) and denote Y t = W θt for all t ≥
0. The family ( Y t ) t ≥ is a collection of positive random variables verifyingsup t ≥ P (cid:2) Y /θt (cid:3) = sup t ≥ P [ W t ] = 1 < ∞ . As 1 /θ is strictly greater than 1, this relation implies the uniform integrablity of ( Y t ) t ≥ . Sincethe process ( W θt ) t ≥ converges almost surely to W θ ∞ , we get from uniform integrability thatlim t →∞ P h W θt i = P h W θ ∞ i . (3.1.9)Now, one can observe that the right hand side term is positive if and only if (3.1.3) holds andthat it is zero if and only if (3.1.4) holds. To prove the theorem, it is then enough to provethat β P (cid:2) W θ ∞ (cid:3) is a non-increasing function of | β | and choose for example¯ β + c = inf { β ≥ P h W θ ∞ i = 0 } , (3.1.10)which does not depend on θ ∈ (0 , β P (cid:2) W θt (cid:3) is an non-increasing function of | β | for all positive t . By standard arguments, we get that ∂∂β P h W θt i = P (cid:20) θW θ − t ∂∂β W t (cid:21) = P h θW θ − t P h(cid:16) ω ( V t ) − λ ′ ( β ) νr d t (cid:17) e βω ( V t ) − λ ( β ) νr d t ii = θ P P h W θ − t (cid:16) ω ( V t ) − λ ′ ( β ) νr d t (cid:17) e βω ( V t ) − λ ( β ) νr d t i . Introducing the probability measure P β on point measures, given byd P β ( ω ) = e βω ( V t ) − λr d νt d P ( ω ) , the derivative of P [ W θt ] is now given by ∂∂β P h W θt i = θ P P β h W θ − t ( ω ( V t ) − r d νλ ′ t ) i . (3.1.11) PPE P β , ω is aPoisson point process on R + × R .We can then use the Harris-FKG inequality for Poisson processes [40, th. 11 p. 31] in orderto bound the above expectation. Indeed, the variable ω ( V t ) − r d νλ ′ t is an increasing function ofthe point process and by definition, the process W θ − t is then a decreasing function of ω when β ≥ β < β P β h W θ − t ( ω ( V t ) − λ ′ r d νt ) i ≤ P β h W θ − t i P β h ( ω ( V t ) − r d νλ ′ t ) i = 0 , (3.1.12)where the last equality is a result of the relation P [ ω ( V t ) e βω ( V t ) ] = λ ′ ( β ) r d νt. The same result with opposite inequality comes when β <
0. Thus, we get from (3.1.11) and(3.1.12) that P [ W θt ] is a non-increasing function of | β | .We recall at this point that Poisson processes with mutually absolutely continuous intensitymeasures are themselves mutually absolutely continuous. Proposition 3.1.4.
Let η be a Poisson point process on a measurable space E , of intensitymeasure µ . Let f be a function such that e f − ∈ L ( µ ) . Then, under the probability measure Q defined by d Q d P = exp (cid:18)Z E f ( x ) η (d x ) − Z E ( e f ( x ) −
1) d µ ( x ) (cid:19) , the process η is a Poisson point process of intensity measure e f d µ .Proof. Let g be any non-negative measurable function. As the Laplace functional characterizesPoisson processes (theorem 3.9 in [41]), we compute it for the point process η under the measure Q : Q exp (cid:26) − Z E g ( s ) η (d x ) (cid:27) = P exp (cid:26)Z E f ( x ) − g ( x ) η (d x ) (cid:27) e − R ( e f ( x ) −
1) d µ ( x ) = exp (cid:26)Z E ( e f ( x ) − g ( x ) −
1) d µ ( x ) (cid:27) e − R ( e f ( x ) −
1) d µ ( x ) = exp (cid:26)Z E ( e − g ( x ) − e f ( x ) d µ ( x ) (cid:27) , where the second equality is an application of (2.2.7). The expression we obtain corresponds,as claimed, to a Poisson point process of intensity measure e f d µ . W ∞ In this section, we prove that one can take the limit in the identity W s + t = P [ e t W s ◦ θ t,B t ] andobtain the equation of self-consistency: W ∞ = P [ e t W ∞ ◦ θ t,B t ] . (3.2.13) PPE d = 1.A part of the problem is that we only have almost sure convergence of the W s ◦ θ t,x for countablenumber of x ’s. To deal with this issue, we show that the quantity W t ( x ) := e − λ ( β ) νr d t P x [exp ( βω ( V t ))] , (3.2.14)does not vary too much with x , in the sense of the following lemma: Lemma 3.2.1.
There exists a constant C = C ( β, ν, r ) , such that, for all t ∈ [0 , ∞ ] and x, y ∈ R d , P [ | W t ( x ) − W t ( y ) | ] ≤ C | x − y | . (3.2.15) Proof.
To simplify the notations, we only consider the case where y = 0. To recover the lemma,it is enough to argue that the Poisson environment is invariant in law under a translation inspace of vector y .To prove (3.2.15) for y = 0, we write the difference of the two martingales as expectationsover two coupled Brownian motions, in the same environment. The coupling we consider is themirror coupling, which is defined as follows (see [30] for more details).At time 0, one of the Brownian motions is starting at 0 and one is starting at x ∈ R d .Denote by H be the hyperplane bisecting the segment [0 , x ], which is the hyperplane passingby x/ x . Let also τ = inf { t ≥ | B t ∈ H } , be the first hitting time of H by B . Then, define e B as the path that coincides with the reflectionof path of B with respect to H for times before τ , and that coincides with B after τ . PPE e B has the law of a Brownian motion starting from x . Moreover, the time τ isthe first time B and e B meet. After τ , the processes coincide. The variable τ has the followingcumulative distribution function: P ( τ ≥ z ) = φ z ( | x | ) , (3.2.16)where, for positive z , φ z ( | x | ) = 2 √ πz Z | x | / e − u / z d u, and where | · | is the Euclidean distance. In the litterature, a coupling that satisfies this relationis said to be maximal (see [30]). Let e t = exp( βω ( V t ( B )) − λ ( β ) νr d t ) , ˜ e t = exp( βω ( V t ( e B )) − λ ( β ) νr d t ) , which we factorize in the contributions before and after B and e B coalesce, so that, for t ∈ [0 , ∞ ), P [ | W t ( x ) − W t (0) | ] ≤ P P [ | ˜ e t − e t | ]= P P [ | (˜ e t ∧ τ − e t ∧ τ ) e ( t − t ∧ τ ) + ◦ θ t ∧ τ,B t ∧ τ | ]= P P [ | ˜ e t ∧ τ − e t ∧ τ | ] , where the last equality is a result of the independance of the Poisson environment before andstrictly after time t ∧ τ .Then, we distinguish the cases where B or e B encounter a point of the environment before t ∧ τ , and the cases where they don’t. We get that P P [ | ˜ e t ∧ τ − e t ∧ τ | ] writes: P P h | e t ∧ τ − ˜ e t ∧ τ | { ω ( V t ∧ τ ( B )) > , ω ( V t ∧ τ ( e B )) > } i + P P h ( e t ∧ τ − e − λνr d t ∧ τ ) { ω ( V t ∧ τ ( B )) > , ω ( V t ∧ τ ( e B ))=0 } i + P P h (˜ e t ∧ τ − e − λνr d t ∧ τ ) { ω ( V t ∧ τ ( B )=0 , ω ( V t ∧ τ ( e B ) > } i . We first use the triangle inequality in the first expectation of the sum, and neglect the negativeterms in two other expectations. Then, recombining the terms, one obtains that P [ | W t ( x ) − W t (0) | ] ≤ P P (cid:2) e t ∧ τ { ω ( V t ∧ τ ( B )) > } (cid:3) + 2 P P h ˜ e t ∧ τ { ω ( V t ∧ τ ( e B )) > } i = 4 P P (cid:2) e t ∧ τ { ω ( V t ∧ τ ( B )) > } (cid:3) ≤ P P (cid:2) e τ { ω ( V τ ( B )) > } (cid:3) , (3.2.17)where the equality is a consequence of invariance in law of the Poisson environment under thetranslation by x .For any fixed τ , the variable ω ( V τ ) is a Poisson r.v. of parameter νr d τ , so that P P (cid:2) e τ { ω ( V τ ( B ) > } (cid:3) = ∞ X k =1 P P [ e τ { ω ( V τ )= k } ] = ∞ X k =1 P (cid:20) e βk − λνr d τ ( νr d τ ) k k ! e − νr d τ (cid:21) , from which we get by standard computations that P P (cid:2) e τ { ω ( V τ ( B ) > } (cid:3) = P [1 − exp( − e β νr d τ )] . (3.2.18)To control this last expectation, we will first notice that τ / | x | is independent of | x | , andwe will compute its density. By equation (3.2.16) and the change of variable u = | x | p ( z ) v , weget that P ( τ / | x | > z ) = 1 √ π Z / √ z e − v / d v. PPE z is continuous and everywhere differentiable on [0 , ∞ ). The variable τ / | x | hence admits a density with respect to the Lebesgue measure, given by f ( z ) = 14 √ π e − / z z / . Therefore, one gets that the right-hand side of (3.2.18) can be written as P h(cid:16) − exp( − e β νr d τ | x | − | x | ) (cid:17) { τ/ | x | ≤ } i + P h(cid:16) − exp( − e β νr d τ | x | − | x | ) (cid:17) { τ/ | x | > } i ≤ e β νr d | x | + Z ∞ (cid:16) − exp( − e β νr d z | x | ) (cid:17) f ( z )d z. As there is some constant
C > f ( z ) ≤ Cz − / , after the change of variables z | x | = u , the integral above can be bounded by | x | Z ∞ C (cid:16) − exp (cid:16) − e β νr d u (cid:17)(cid:17) v − / d v, where one can check that the integral converges. Since its value only depends on β, ν and r ,we finally get that there exists some constant C ′ = C ( β, ν, r ), such that P [ | W t ( x ) − W t (0) | ] ≤ P [1 − exp( − e β νr d τ )] ≤ C ′ ( | x | + | x | ) , where the first inequality comes from combining (3.2.17) and (3.2.18). Since the W t ( x ) variableshave bounded expectations, this proves the lemma in the case where t < ∞ . The t = ∞ caseis then a consequence of Fatou’s lemma.We can now show that the self-consistency equation holds. Let δ > q ∈ Z d , define ∆( q ) to be the cube of length δ centered at δq , so that allthe cubes form a partition of the space R d . We get that the right-hand side of (3.1.6) satisfies P [ e t W s ◦ θ t,B t ] = X q ∈ Z d P [ e t W s ◦ θ t,B t ; B t ∈ ∆( q )]= X q ∈ Z d P [ e t W s ◦ θ t,δq ; B t ∈ ∆( q )] + A δs , (3.2.19)where A δs = P q ∈ Z d P [ e t ( W s ◦ θ t,B t − W s ◦ θ t,δq ); B t ∈ ∆( q )] . First observe that P -almost surely, W s ◦ θ t,δq converges to W ∞ ◦ θ t,δq for all q ∈ Z d , soFatou’s lemma entails P -a.s. , lim inf s →∞ X q ∈ Z d P [ e t W s ◦ θ t,δq ; B t ∈ ∆( q )] ≥ X q ∈ Z d P [ e t W ∞ ◦ θ t,δq ; B t ∈ ∆( q )] , so that, by (3.2.19) and letting s → ∞ in (3.1.6), P -a.s. , W ∞ ≥ X q ∈ Z d P [ e t W ∞ ◦ θ t,δq ; B t ∈ ∆( q )] + lim inf s →∞ A δs . (3.2.20)Furthermore, using the fact that W is a martingale, one can check that s A δs is also amartingale with respect to the filtration {G t + s } s ≥ . For any time S ≥
0, Lemma 3.2.1 implies
PPE P [ | A δS | ] ≤ X q ∈ Z d P (cid:2) P [ e t | W S ◦ θ t,B t − W S ◦ θ t,δq | ] B t ∈ ∆( q ) (cid:3) ≤ C ( δ + δ ) X q ∈ Z d P (cid:2) B t ∈ ∆( q ) (cid:3) = C ( δ + δ ) , where, in the second inequality, we have factorized by P [ e t ] = 1, using the independence under P of the environment before and strictly after time t . Thus, by Doob’s inequality [35, Th. 3.8(i), Ch. 1], we have for all u > P " sup ≤ s ≤ S | A δs | ≥ u ≤ C ( δ + δ ) u , where we can let S → ∞ by monotone convergence. This implies that sup s ≥ | A δs | converges inprobability to 0, when δ →
0, which in turn implies thatlim inf s →∞ | A δs | P −→ . Then, using Lemma 3.2.1 in the case where t = ∞ , the same computation as above wouldshow that P (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X q ∈ Z d P [ e t W ∞ ◦ θ t,B t ; B t ∈ ∆( q )] − X q ∈ Z d P [ e t W ∞ ◦ θ t,δq ; B t ∈ ∆( q )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( δ + δ ) , and hence, X q ∈ Z d P [ e t W ∞ ◦ θ t,δq ; B t ∈ ∆( q )] L −→ P [ e t W ∞ ◦ θ t,B t ] . In particular, we have shown that the right-hand side of (3.2.20) converges in probability to P [ e t W ∞ ◦ θ t,B t ], so that almost surely W ∞ ≥ P [ e t W ∞ ◦ θ t,B t ] . Observe that the two quantities have the same expectations to conclude that (3.2.13) holds.
The martingale W t is uniformly integrable if and only if the polymer is inthe weak disorder phase, i.e. W ∞ > , P -almost surely.Proof. If W t is UI, then W t converges in L to W ∞ , so that P [ W ∞ ] = 1 >
0, therefore weakdisorder must hold by the dichotomy.Suppose now that the polymer is in weak disorder and set X t,x = W ∞ ◦ θ t,x P [ W ∞ ] . The self-consistency equation (3.2.13) writes X , = P [ e t X t,B t ], so that, as X t,x is independentof G t for all x , P [ X , |G t ] = P [ e t P [ X t,B t ]] = P [ e t ] . This shows that for all t ≥ W t = P [ X , |G t ] a.s.Hence, ( W t ) t ≥ is uniformly integrable since the family of the right-hand side is a uniformlyintegrable martingale. PPE -regionTheorem 3.3.1. (i) There exist two critical parameters β − ∈ [ −∞ , and β ∈ [0 , ∞ ] ,depending only on ν, r and d , such that, if β ∈ ( β − , β ) ∪ { } then sup t ∈ R P [ W t ] < ∞ , (3.3.21) and such that the supremum is infinite if β ∈ R \ [ β − , β +2 ] .(ii) Furthermore, if d ≥ , there exists a constant c ( d ) ∈ (0 , ∞ ) , such that (3.3.21) holdswhenever λ ( β ) νr d +2 < c ( d ) , (3.3.22) (iii) In particular, β − < and β > whenever d ≥ .(iv) Also for d ≥ , when νr d +2 < c ( d ) the constant in (3.3.22) , we have β − = −∞ . Definition 3.3.2.
We call the L -region the set of parameters β, ν, r for which (3.3.21) holds.Proof of Theorem 3.3.1. We introduce the product measure P ⊗ of two independent Brownianmotions B t and ˜ B t starting from 0 with respective tubes V t and ˜ V t . The main idea is to write W t = P ⊗ [ e βω ( V t ) e βω ( ˜ V t ) ] e − λνr d t , so that, using Fubini’s theorem, P [ W t ] = P ⊗ P [ e β ( ω ( V t )+ ω ( ˜ V t ) ] e − λνr d t . (3.3.23)One can see that ω ( V t ) + ω ( ˜ V t ) = 2 ω ( V t ∩ ˜ V t ) + ω ( V t ∆ ˜ V t ) , which is the sum of two independent Poisson random variables; computing their Laplace trans-forms leads us to P [ W t ] = exp (cid:16) λ (2 β ) ν | V t ∩ ˜ V t | + λν | V t ∆ ˜ V t | − λνr d t (cid:17) = exp (cid:16) λ ν | V t ∩ ˜ V t | (cid:17) −→ t →∞ exp (cid:16) λ ν | V ∞ ∩ ˜ V ∞ | (cid:17) , (3.3.24)where the second equality is obtained using | V t | = | ˜ V t | = tr d and λ ( β ) = λ (2 β ) − λ ( β ), whilethe limit is justified by monotone convergence. Now, | V ∞ ∩ ˜ V ∞ | = Z ∞ | U ( B t ) ∩ U ( ˜ B t ) | d t = Z ∞ | U (0) ∩ U ( ˜ B t − B t ) | d t law = Z ∞ | U (0) ∩ U ( B t ) | d t, since ( B t + ˜ B t ) t ≥ = ( B t ) t ≥ . Hence, using monotone convergence and (3.3.24), we get thatsup t ∈ R P [ W t ] = lim t →∞ P [ W t ] = P (cid:20) exp (cid:18) λ ( β ) ν Z ∞ | U (0) ∩ U ( B t ) | d t (cid:19)(cid:21) , (3.3.25) PPE W t ) t ≥ being a submartingale. Equation (3.3.25)shows that sup t P [ W t ] is an increasing function of | β | , which proves part (i) of the theorem.To prove the second part, we will bound the right hand side of (3.3.25). First observe that | U (0) ∩ U ( B t ) | ≤ | U (0) | | B t |≤ γ − /dd r law = r d ((cid:12)(cid:12)(cid:12)(cid:12)(cid:12) B tγ /dd r !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ) , so that, by a change of variables,sup t ∈ R P [ W t ] ≤ P (cid:20) exp (cid:18) λ ( β ) νr d γ − /dd Z ∞ | B t |≤ d t (cid:19)(cid:21) . (3.3.26)When d ≥
3, the Brownian motion is transient and has the following property: α ( d ) = sup x ∈ R d P x (cid:20)Z ∞ | B t |≤ d t (cid:21) < ∞ . By Khas’minskii’s lemma [61, p. 8, Lemma 2.1], this implies that P (cid:20) exp (cid:18) u Z ∞ | B t |≤ d t (cid:19)(cid:21) < (1 − uα ) − , whenever uα <
1. Looking back at (3.3.26), this condition finally leads to (3.3.22).Part (iii) is obtained by observing that λ ( β ) → β →
0, so that condition (3.3.22) isfulfilled for small enough β .To prove (iv), just note that the right-hand side of (3.3.22) tends to νr d +2 as β → −∞ . The critical values β ± , ¯ β ± c and β ± c defined in Theorems 2.5.1, 3.1.3 and 3.3.1 are ordered. Proposition 3.4.1.
The following properties hold:(i) For all d ≥ , β +2 ≤ ¯ β + c ≤ β + c < ∞ and β − c ≤ ¯ β − c ≤ β − . (3.4.27) (ii) When d ≥ , these parameters are all non-zero.(iii) When d ≥ and νr d +2 is small enough, β − c = ¯ β − c = β − = −∞ . Remark 3.4.2.
Point (iii) tells us that if the intensity of the Poisson point process or theradius are sufficiently small, the polymer will not really be impacted by the environment.
Remark 3.4.3.
For d = 1 , , it holds that that β ± = ¯ β ± c = β ± c = 0 . The reader is referred tothe proofs in [8, 10, 19, 39] for other similar models, and can be convinced that the argumentsgo through in our case [9]. Remark 3.4.4.
A long-standing conjecture is that β ± c = ¯ β ± c , i.e., that the weak/strong disordertransition coincide with the high/low temperature one. PPE Proof.
We first show that β +2 ≤ ¯ β + c . Suppose β +2 > ≤ β < β +2 . By definition, wehave sup t ≥ P [ W t ] < ∞ , so that ( W t ) t ≥ is a martingale bounded in L . Thus, ( W t ) t ≥ converges in L norm, whichimplies L convergence.Since P [ W t ] = 1 for all t , we get that P [ W ∞ ] = 1, so (3.1.3) must hold and hence β ≤ ¯ β + c .As it is true for all β < β +2 , the desired inequality follows directly.We now turn to the proof of ¯ β + c ≤ β + c . Again, suppose that ¯ β + c > ≤ β < ¯ β + c .We have W ∞ > W t → ln W ∞ almost surely, that is to sayln Z t − λνr d t −→ t →∞ ln W ∞ . Dividing by t , we get that 1 t ln Z t − λνr d −→ t →∞ , thus p ( β, ν, r ) = λνr d , i.e. β ≤ β c . The same argument goes for the negative critical values.This ends the proof of (i).Points (ii) and (iii) are repeated from Theorem 3.3.1. Directional free energy
In this section we make use of the Brownian nature of the polymer and the invariance ofthe medium under shear transformations, which induces a lot of symmetries in the model,culminating with quadratic shape function and the equality (4.2.11).
With P t,ys,x the Brownian bridge in R d joining ( s, x ) to ( t, y ), we introduce the point-to-pointpartition (P2P) function Z t ( ω, β ; x ) = P t,x , [exp { βω ( V t ) } ] , (4.1.1)from which we can recover the point-to-level (P2L) partition function Z t ( ω, β ) = Z R d Z t ( ω, β ; x ) ρ ( t, x ) dx (4.1.2)by conditioning on B t . We use the standard notation ρ ( t, x ) = (2 πt ) − d/ exp −| x | / t for theheat kernel in R d . For ξ ∈ R d , define the shear transformation τ ξ : R + × R d → R + × R d by τ ξ ( s, x ) = ( s, x + sξ ) , which is one to one with τ − ξ = τ − ξ . Since τ ξ acts on the graph of functions f : R + → R d , wedenote its action on functions by τ ξ f : s f ( s ) + sξ , (4.1.3)so that τ ξ ( s, f ( s )) = ( s, τ ξ ( f )( s ). The pushed forward of a point measure by τ ξ is defined by τ ξ ◦ (cid:0) X i δ ( t i ,x i ) (cid:1) = X i δ ( t i ,x i + ξt i ) = X i δ τ ξ ( t i ,x i ) , (4.1.4)where it is clear that τ ξ ◦ ω is again a Poisson point process with intensity νdsdx , i.e., τ ξ ◦ ω = ω in law. With B the canonical process, under the measure P t,tξ , the process W = τ − ξ ( B ) is aBrownian bridge (0 , → (0 , ω , Z t ( ω, β ; tξ ) = P t,tξ , [exp { βω ( V t ( B )) } ]= P t,tξ , [exp { βω ( V t ( τ ξ ( W ))) } ]= P , , [exp { βω ( τ ξ ( V t ( B ))) } ]= P , , [exp { β ( τ ξ ◦ ω )( V t ( B )) } ]= Z t ( τ ξ ◦ ω, β ; 0) . (4.1.5)This implies that Z t ( ω, β ; x ) has same law as Z t ( ω, β ; 0). We can prove that the directional freeenergy , in the direction ξ ∈ R d , p dir ( β, ν ; ξ ) = lim t →∞ t ln Z t ( ω, β ; tξ )exists a.s. and in L p -norm for all p ≥
1, and is equal to lim t →∞ t − P [ln Z t ( ω, β ; tξ )]. The routeis quite different from Theorem 2.3.1, it follows the lines of chapter 5 in [61] for the undirected20 PPE s ≤ t , V s,t = V s,t ( B ) = ∪ u ∈ [ s,t ] { u } × U ( B u ), and also e s,t ( x, y ; ω ) := P h e βω ( V s,t ) B t ∈ U ( y ) (cid:12)(cid:12) B s = x i = P x h e βθ s,x ◦ ω ( V t − s ) B t − s ∈ U ( y ) i , which is the integral of the P2P partition function over a ball of radius r . Then, the quantity a s,t ( x, y ; ω ) = inf z ∈ U ( x ) ln e s,t ( z, y ; ω )is superadditive, in particular we have a ,s + t (0 , ( s + t ) ξ ; ω ) ≥ a ,s (0 , sξ ; ω ) + a s,s + t ( sξ, ( s + t ) ξ ; ω ) , and the subadditive ergodic theorem shows the existence of the limit t − a ,t (0 , tξ ; ω ) as t → ∞ ,say p dir ( β, ν ; ξ ), a.s. and in L . ( L p -convergence will follow from the concentration inequality,which remains unchanged). Then, one can show that the infimum over z ∈ U ( x ) in thedefinition of a ,t (0 , y ; ω ) can be dropped in the limit t → ∞ , as well as the integration in thedefinition of e ,t (0 , y ; ω ) on the fixed domain U ( y ). Proving these claims requires some workwith quite a few technical estimates; we do not write the details here, the reader is referred tosection 5.1 in [61].By (4.1.5), Z t ( ω, β ; x ) law = Z t ( ω, β ; x ′ ) and thus, for all ξ, ξ ′ ∈ R d , p dir ( β, ν ; ξ ) = p dir ( β, ν ; ξ ′ ) . (4.1.6) Let P h be the Wiener measure with drift h ∈ R d , i.e., the probability measure on the pathspace C ( R + , R d ) such that for all t , (cid:18) dP h dP (cid:19) |F t = exp (cid:26) h · B t − t | h | (cid:27) . By Cameron-Martin formula, under P h , the canonical process B is a Brownian motion withdrift h , i.e., W = τ − h ( B ) is a standard Brownian motion under P h and has the same law as B under P . Thus, the partition function for the drifted Brownian polymer Z ht ( ω, β ) def . = P h [exp { βω ( V t ( B )) } ]= P h [exp { βω ( V t ( τ h ( W ))) } ]= P [exp { βω ( V t ( τ h ( B ))) } ]= Z t ( τ − h ◦ ω, β ) (4.2.7)as in (4.1.5). It would be routine, and this time exactly as in the proof of Theorem 2.3.1, toshow the existence of free energy for the drifted Brownian polymer p h ( β, ν ) := lim t →∞ t − ln Z ht ( ω, β )a.s. and in L p ; But in fact, this is even unnecessary since (4.2.7) yields the existence of thelimit. Now, the previous display together with invariance of ω under shear shifts imply that p h ( β, ν ) = p ( β, ν ) . (4.2.8) PPE Z ht ( ω, β ) = Z R d e h · x − t | h | / Z t ( ω, β ; x ) ρ ( t, x ) dx . (4.2.9)By Laplace method, it follows from standard work that p h ( β, ν ) = sup ξ ∈ R d n h · ξ − | h | / − | ξ | / p dir ( β, ν ; ξ ) o (4.1.6) = p dir ( β, ν ; 0) − | h | / ξ ∈ R d (cid:8) h · ξ − | ξ | / (cid:9) (4.2.10)= p dir ( β, ν ; 0) . Finally, all the above notions of free energy coincide: p ( β, ν ) = p dir ( β, ν ; ξ ) = p h ( β, ν ) . (4.2.11) Conclusion:
The critical values β ± c for equality of quenched and annealed free energy are thesame for all free energies (P2P in all directions, P2L with all drifts). In the L -region, a local limit theorem was discovered by Sina¨ı [59] in the discrete case, andextended to our continuous model by Vargas [63].Define the time-space reversal operator on the environment θ ← t,x , acting on point measuresas θ ← t,x ( X i δ ( t i ,x i ) (cid:1) = X i δ ( t − t i ,x i − x ) (4.3.12) Theorem 4.3.1 (Local limit theorem; [63], Th. 2.9) . Assume β ∈ ( − β − , β +2 ) . Then, for anyconstant A > and any positive function ℓ t tending to ∞ with ℓ t = o ( t a ) for some a < / , Z t ( ω, β ; x ) = W ∞ × W ∞ ◦ θ ← t,x + ε t ( x ) , (4.3.13) and Z t ( ω, β ; x ) = W ℓ t × W ℓ t ◦ θ ← t,x + δ t ( x ) , with error terms vanishing as t → ∞ , sup | x |≤ A √ t P [ | ε t ( x ) | ] → , sup | x |≤ A √ t P [ | δ t ( x ) | ] → . Intuitively, the local limit theorem states that, the polymer ending at x at time t only”feels” the environment at times s close to 0 and locations close to 0 or close t at ”large” times s close to t and locations close to x . (See Figure 4.1.) In between, it behaves like a Brownianbridge. Conjecture 4.3.2.
We formulate two conjectures: • It is natural to define another pair of critical inverse temperature, analogue to the weak/strongdisorder transition: β + ,dirc = sup { β ≥ t P [( W dir,ξt ) / ] > } , (4.3.14) β − ,dirc = inf { β ≤ t P [( W dir,ξt ) / ] > } . PPE Using Jensen inequality in (4.1.2) , it is not difficult to get β + ,dirc ∈ [ β +2 , β + c ] , β − ,dirc ∈ [ β − c , β − ] .We conjecture that the equality holds, i.e., β ± c = β ± ,dirc . • A long standing conjecture is that the local limit theorem (4.3.13) holds the way all throughthe weak disorder region. Note that the latter conjecture would imply that the former one holds.
Figure 4.1: The local limit theorem. The P2P partition function only feels details of theenvironment close to the space-time endpoints (0 ,
0) and ( t, x ). In between, it behaves like theGaussian propagator.
The replica overlap and localization
The utlimate goal of this section is to show that the thermodynamic phase transition of section2 is a localization transition for the polymer.To do that, we need fine tools from stochastic analysis. The starting point is Doob-Meyerdecomposition, a natural and strong tool to study stochastic processes in which the processis written as the sum of a local martingale (the impredictable part) and a bounded variationpredictable process (the tamed part). We start by recalling some martingale properties of thePoisson environment that will prove usefull throughout the following chapters.
Given the Poisson point process ω , we introduce the compensated measure ¯ ω ,¯ ω (d s d x ) = ω (d s d x ) − ν d s d x, (5.1.1)and we abreviate its restriction to (0 , t ] × R d by ¯ ω t . By definition, for all function f ( s, x, ω )that verifies Z [0 ,t ] × R d P [ | f ( s, x, · ) | ]d s d x < ∞ , (5.1.2)the compensated integral of f is given by Z f ( s, x, ω )¯ ω t (d s d x ) = Z f ( s, x, ω ) ω t (d s d x ) − Z [0 ,t ] × R d f ( s, x, ω ) ν d s d x. (5.1.3)Furthermore, we say that a function f ( t, x, ω ) is predictable , if it belongs to the sigma-fieldgenerated by all the functions g ( t, x, ω ) that satisfy the following properties:(i) for all t >
0, ( x, ω ) → g ( t, x, ω ) is B ( R d ) × G t -measurable;(ii) for all ( x, ω ), t → g ( t, x, ω ) is left continuous.Then, if the function f ( t, x, ω ) is predictable, provided that (5.1.2) holds and that Z [0 ,t ] × R d P [ f ( s, x, · ) ]d s d x < ∞ , the process t → R f d¯ ω t is a square-integrable martingale associated to ( G t ) t ≥ , of previsiblebracket [31, Section II.3.] (cid:28)Z f d¯ ω (cid:29) t = Z [0 ,t ] × R d f ν d s d x. (5.1.4)The previsible bracket has the property that ( R f d¯ ω t ) −h R f d¯ ω i t is a martingale. In particular,Var (cid:18)Z f ( s, x, · ) ω t (d s d x ) (cid:19) = Z [0 ,t ] × R d P [ f ( s, x, · ) ] ν d s d x. (5.1.5)24 PPE ln Z t Since W t is a martingale and ln is concave, − ln( W t ) is a submartingale, for which we want toget a Doob-Meyer decomposition [53, Ch.VI].In the following, we will use the notation ∆ s X := X s − X s − for any c`adl`ag process X .Let ζ t := e βω ( V t ) . As ω ( V t ) can be expressed as a sum over ω t , we get by telescopic sum that ζ t = 1 + R ω t (d s d x )∆ s ζ , an integral over R + × R d . Averaging over the Brownian path, we alsoget that Z t can be expressed as a sum over the process. Therefore, by telescopic sum,ln Z t = Z ω t (d s d x )∆ s ln Z. Now, let ( t, x ) ∈ R + × R d be any point of the point process ω . As ( t, x ) is almost surely theonly point of ω at time t , we can write that Z t = P h e β e βω ( V t − ) χ t,x ( B ) i + P h e βω ( V t − ) (1 − χ t,x ( B )) i = P h ( e β − e βω ( V t − ) χ t,x ( B ) i + P h e βω ( V t − ) i = (cid:16) λ ( β ) P β,ωt − [ χ t,x ] + 1 (cid:17) Z t − . Hence, ln Z t = Z ω t (d s d x ) ln (cid:0) λP β,ωs − [ χ s,x ] (cid:1) . (5.2.6)Let g be the function g ( u ) = u − ln(1 + u ), which is positive on ( − , ∞ ). Then, recalling that R R d χ s,x dx = r d , we obtain Doob’s decomposition − ln W t = λνr d t − ln Z t = M t + A t , (5.2.7)where the martingale M and the increasing process A are given by M t = − Z ¯ ω t (d s d x ) ln (cid:0) λP β,ωs − [ χ s,x ] (cid:1) , (5.2.8) A t = λνr d t − Z [0 ,t ] × R d ln (cid:0) λP β,ωs − [ χ s,x ] (cid:1) ν d s d x = Z [0 ,t ] × R d g (cid:0) λP β,ωs − [ χ s,x ] (cid:1) ν d s d x. (5.2.9)Moreover, the martingale M is square-integrable, and its bracket is given by h M i t = Z [0 ,t ] × R d (cid:2) ln (cid:0) λP β,ωs − [ χ s,x ] (cid:1)(cid:3) ν d s d x. Things being clear from the context, we will use the same notation | A | to denote the Lebesguemeasure of Borel subset A of R d or R + × R d . Definition 5.3.1.
For any two paths B and ˜ B , we define the replica overlap R t ( B, ˜ B ) asthe mean volume overlap of the two tubes around B and ˜ B in time [0 , t ] : R t ( B, ˜ B ) = 1 tr d | V t ( B ) ∩ V t ( ˜ B ) | . (5.3.10) PPE In a similar way, we define the quenched overlaps I t and J t as I t = 1 r d P β,ωt ⊗ (cid:2) | U ( B t ) ∩ U ( ˜ B t ) | (cid:3) , (5.3.11) J t = 1 tr d P β,ωt ⊗ (cid:2) | V t ( B ) ∩ V t ( ˜ B ) | (cid:3) , (5.3.12)The variable I t stands for the expected volume of overlap around the endpoints of twoindependent polymer paths, while J t is the expected volume of overlap during the time interval[0 , t ]. Note that both R t I s d s and tJ t represent an expected volume of overlap in time [0 , t ], butthey will emerge from different circumstances. Similar to χ s,x = { x ∈ U ( B s ) } from definition(2.1.3), we write for short ˜ χ s,x = { x ∈ U ( ˜ B s ) } . Writing | U ( B t ) ∩ U ( ˜ B t ) | = Z R d χ t,x ˜ χ t,x dx, we derive two useful formulas: I t = 1 r d Z R d P β,ωt ( χ t,x ) dx , J t = 1 r d Z R d dx t Z t P β,ωt ( χ s,x ) ds . (5.3.13)For better comparisons, we have normalized all quantities in (5.3.10), (5.3.11) and (5.3.12)in such a way that 0 ≤ R t , I t , J t ≤ , so that we can – and we will – view each of them as a localization index : • R t close to 1 means that the two fixed paths B, ˜ B are close on the interval [0 , t ]; • I t close to 1 means that the endpoints of two independent samples of the polymer measureare typically close one from the other; • J t close to 1 means that the paths of two independent polymers are close all along thetime interval.The second case corresponds to endpoint localization whereas the third one is path localization .Mathematically, the quantity I t appears via Itˆo’s calculus (stochastic differentiation) and J t via Malliavin calculus (integration by parts). On the contrary, small values of these indicescorrespond to absence of localization: it means that the polymer spreads more or less uniformlyin space without particular preference. Remark 5.3.2.
When β = 0 , the Gibbs measure P β,ωt reduces to Wiener measure P , so that J t = R t I s d s = tP ⊗ [ R t ( B, ˜ B )] , and P ⊗ [ R t ( B, ˜ B )] = 1 t Z t P ⊗ (cid:2) | U ( B s ) ∩ U ( ˜ B s ) | (cid:3) d s ≤ r d t Z t P ( B s ∈ U (0)) d s −→ t → . Thus, lim t →∞ J t = lim t →∞ t Z t I s d s = 0 , This indicates that, in absence of interaction with the inhomogeneous medium, there is nolocalization of the Brownian path.
We now come to the core of the section: localization results. The product measure P β,ωt ⊗ = P β,ωt ⊗ P β,ωt makes the 2 replicas B, ˜ B independent polymer paths sharingthe same environment ω . PPE A naive prediction is that, for small β , the polymer is a small perturbation of Brownian motionfor small β , with a comparable behavior, whereas for large β localization takes place and thelimits are nonzero. A first theorem shows that this is the case for the quantity I t : Theorem 5.4.1.
The following equivalence holds for β = 0 : W ∞ = 0 ⇐⇒ Z ∞ I s d s = ∞ , P -a.s. (5.4.14) In particular, the above integral is a.s. finite for β ∈ ( ¯ β − c , ¯ β + c ) , and a.s. infinite for β < ¯ β − c or β > ¯ β + c . Moreover, we have: lim t →∞ t Z t I s d s = 0 if β ∈ [ β − c , β + c ] ∩ R , (5.4.15)lim inf t →∞ t Z t I s d s > β ∈ R \ [ β − c , β + c ] . (5.4.16) Remark 5.4.2.
Note that R ∞ I s d s is a.s. finite or a.s. infinite, by the dichotomy betweenstrong and weak disorder. Similarly, strict positivity of lim inf t →∞ t − R t I s ds is equivalent tolow temperature region ψ ( β, ν ) > in (2.5.20) . In fact, we will prove in (5.4.19) – (5.4.23) thatunder strong disorder, there exist c , c ∈ (0 , ∞ ) , such that c Z t I s d s ≤ − ln W t ≤ c Z t I s d s, for large t , P − a.s. (5.4.17) Proof. (Theorem 5.4.1) First observe that one can easily derive (5.4.15) and (5.4.16) from(5.4.14) and (5.4.17).To show (5.4.17), we will relate R t I s d s to the variables M t and A t of the Doob decomposition(5.2.7) as follows. From (5.3.13), we have Z [0 ,t ] × R d (cid:16) P β,ωs [ χ s,x ] (cid:17) d s d x = r d Z t I s d s. (5.4.18)Then, looking at the behavior of g ( u ) in (5.2.9) and ln(1 + u ) around 0, it is clear that thereare two constants c , c >
0, depending only on β and ν , such that c u ≤ νg ( u ) ≤ c u , ν ln(1 + u ) ≤ c u , for all u in [0 , λ ] when β ≥
0, (resp. [ λ,
0] when β ≤ c Z t I s d s ≤ A t ≤ c Z t I s d s, (5.4.19) h M i t ≤ c Z t I s d s. (5.4.20)We then recall two results about martingales. (These facts for the discrete martingales arestandard (e.g. [26, p. 255, (4.9),(4.10)]. It is not difficult to adapt the proof for the discretesetting to our case.) Let ε >
0, then h M i ∞ < ∞ = ⇒ ( M t ) t ≥ converges a.s. , (5.4.21) h M i ∞ = ∞ = ⇒ lim t →∞ M t / h M i ε t = 0 a . s .. (5.4.22) PPE P -almost surely, Z ∞ I s d s < ∞ ⇐⇒ A ∞ < ∞ , h M i ∞ < ∞ = ⇒ A ∞ < ∞ , lim t →∞ M t exists and is finite= ⇒ W ∞ > , where the last implication comes from the Doob decomposition (5.2.7). By contraposition, thisproves the first implication of (5.4.14). The reverse implication will follow from the argumentsbelow.The next step is to show (5.4.17), so we now suppose that we are in the strong disordersetting. Using (5.4.19), we see that it is enough to show thatlim t →∞ − ln W t A t = 1 , P -a.s. (5.4.23)or equivalently by the Doob decomposition, thatlim t →∞ M t A t = 0 . (5.4.24)As we just proved, the strong disorder implies that R ∞ I s d s = ∞ , which in turn impliesthat A ∞ = ∞ by (5.4.19). Thus, in the case that h M i ∞ < ∞ , the martingale M t converges sothat the condition (5.4.24) directly holds. When h M i ∞ = ∞ , this is still true as M t A t = M t h M i t h M i t A t −→ t →∞ , by (5.4.19, 5.4.20, 5.4.22).Finally, what is left to demonstrate is that P -almost surely W ∞ = 0 on the event R ∞ I s d s = ∞ . In fact, we showed that this event implies both the limit (5.4.23) and A ∞ = ∞ , so that W t → Endpoint localization : As in the discrete case, we can interpret the results from thepresent subsection, in terms of localization for the path. Indeed, it is proven in Sect. 8 of [19],that for some constant c , c sup y ∈ R d P β,ωt [ B s ∈ U ( y )] ≤ P β,ωt ⊗ h(cid:12)(cid:12) U ( B s ) \ U ( ˜ B s ) (cid:12)(cid:12)i ≤ sup y ∈ R d P β,ωt [ B s ∈ U ( y )] (5.4.25)(in fact, the inequality on the right is trivial, and the one on the left is the combination of(5.5.42) and (5.5.43)).The maximum appearing in the above bounds should be viewed as the probability of thefavorite location for B s , under the polymer measure P β,ωt ; for s = t , the supremum is calledthe probability of the favorite endpoint , and the maximizing y is the location of the favoriteendpoint . Both Theorem 5.4.1 and Theorem 5.5.2 are precise statements that the polymerlocalizes in the strong disorder regime in a few specific corridors of width O (1), but spreadsout in a diffuse way in the weak disorder regime. If ψ ( β, ν ) >
0, the Cesaro-limit of probabilityof the favorite endpoint is strictly positive.
PPE Recall that the excess free energy ψ from (2.4.15) is the difference of a smooth function and aconvex function. Hence its right-derivative, resp. left-derivative, (cid:18) ∂ψ∂β (cid:19) + ( β, ν ) = lim β ′ ց β ψ ( β ′ , ν ) − ψ ( β, ν ) β ′ − β , (cid:18) ∂ψ∂β (cid:19) − ( β, ν ) = lim β ′ ր β ψ ( β ′ , ν ) − ψ ( β, ν ) β ′ − β , exists for all β and all ν , and satisfy (cid:16) ∂ψ∂β (cid:17) + ( β, ν ) ≤ (cid:16) ∂ψ∂β (cid:17) − ( β, ν ). For the same reason asabove, ψ ( · , ν ) is differentiable except on a set which is at most countable, and we can write (cid:18) ∂ψ∂β (cid:19) + ( β, ν ) = lim β ′ ≥ β ∂ψ∂β ( β ′ , ν ) , where the limit is over differentiability points β ′ tending to β by larger values. A similarstatement holds for the left-derivative. For further use, we note that for all fixed ν , ψ ( · , ν ) isabsolutely continuous again for the same reason as above.Now, we turn to the properties of the replica overlap J t . The key fact is the followingproposition: Proposition 5.5.1.
There exist two constants c , c ∈ (0 , ∞ ) , depending only on β and ν ,such that ∀ β > , c (cid:18) ∂ψ∂β (cid:19) + ≤ lim inf t →∞ P [ J t ] ≤ lim sup t →∞ P [ J t ] ≤ c (cid:18) ∂ψ∂β (cid:19) − , (5.5.26) ∀ β < , − c (cid:18) ∂ψ∂β (cid:19) − ≤ lim inf t →∞ P [ J t ] ≤ lim sup t →∞ P [ J t ] ≤ − c (cid:18) ∂ψ∂β (cid:19) + . (5.5.27) Proof.
It is not difficult to see from the definition that tJ t = RR [0 ,t ] × R d P β,ωt [ χ s,x ] d s d x . Hence,using equation (2.4.19) and the fact that e −| β | ≤ λP β,ωt [ χ s,x ] ≤ e | β | , we obtain that λνe β −| β | t P [ J t ] ≤ t ∂∂β ψ t ( β, ν ) ≤ νλe β + | β | t P [ J t ] . (5.5.28)Moreover, the excess free energy writes ψ ( β, ν ) = νλ ( β ) r d − p ( β, ν ), where p ( β, ν ) is a convexfunction, defined as the limit, for t → ∞ , of the convex functions p t ( β, ν ) = t P [ln Z t ] . Byconvexity properties, we know that (cid:18) ∂p∂β (cid:19) − ≤ lim inf t →∞ ∂p t ∂β ≤ lim sup t →∞ ∂p t ∂β ≤ (cid:18) ∂p∂β (cid:19) + , which in turns implies that (cid:18) ∂ψ∂β (cid:19) + ≤ lim inf t →∞ ∂ψ t ∂β ≤ lim sup t →∞ ∂ψ t ∂β ≤ (cid:18) ∂ψ∂β (cid:19) − . (5.5.29)The proposition is then a consequence of (5.5.28) and these last inequalities.With this proposition, we can give a characterization of the critical values β ± c , in terms ofthe asymptotics of the overlap: PPE Theorem 5.5.2.
For all β ∈ [ β − c , β + c ] ∩ R , lim t →∞ P [ J t ] = 0 . (5.5.30) Furthermore, β + c = sup { β ′ ≥ ∀ β ∈ [0 , β ′ ] , lim t →∞ P [ J t ] = 0 } = inf { β > t →∞ P [ J t ] > } , (5.5.31) β − c = inf { β ′ ≤ ∀ β ∈ [ β ′ , , lim t →∞ P [ J t ] = 0 } = sup { β < t →∞ P [ J t ] > } . (5.5.32) Proof.
We will focus on the β ≥ β ≤ δ + c = sup { β ′ ≥ ∀ β ∈ [0 , β ′ ] , lim t →∞ P [ J t ] = 0 } , so what we need to show in particular is β + c = δ + c .To prove the first claim of the theorem, from which β + c ≤ δ + c follows directly, it is enough,using (5.5.26), to verify that ∀ β ≤ β + c , (cid:18) ∂ψ∂β (cid:19) − ( β, ν ) = 0 . (5.5.33)This property is true when β ∈ [0 , β + c ), as ψ is constant and set to 0 in this interval. To provethat it extends to β + c if β + c < ∞ , observe that ψ is minimal at β + c , so that (cid:18) ∂ψ∂β (cid:19) − ( β + c , ν ) ≤ ≤ (cid:18) ∂ψ∂β (cid:19) + ( β + c , ν ) . As we saw earlier that ∂ψ∂β + ≤ ∂ψ∂β − always holds, we finally get that ∂ψ∂β − ( β + c ) = ∂ψ∂β + ( β + c ) = 0.We now prove β + c ≥ δ + c . Let β > β + c be finite, so that, by definition, ψ ( β, ν ) >
0. As ψ isabsolutely continuous with ψ (0 , ν ) = 0, one can write ψ ( β, ν ) = Z β ∂ψ∂β + ( β ′ , ν )d β ′ > , which implies that there exists some β ′ ≤ β such that ∂ψ∂β + ( β ′ , ν ) >
0. By equation (5.5.26), weget that lim inf t →∞ t P [ J t ( β )] >
0, hence β ≥ δ + c . As it is true for all β > β + c , we obtain that β + c ≥ δ + c . The favorite path.
Let M be the set of integer-valued Radon measures on R + × R d , equippedwith the sigma-field G generated by the variables ω ( A ), A ∈ R + × R d , so that we will consider ω as a process of the probability space ( M , G , P ). It is possible to define, for all fixed timehorizon t >
0, a measurable function Y ( t ) : [0 , t ] × M → R d ( s, ω ) Y ( t ) s , (5.5.34)which satisfies the property that, P -almost surely, ∀ s ∈ [0 , t ] , P β,ωt (cid:16) B s ∈ U (cid:16) Y ( t ) s (cid:17)(cid:17) = max x ∈ R d P β,ωt (cid:0) B s ∈ U ( x ) (cid:1) . (5.5.35) PPE Y ( t ) .Here, the path s → Y ( t ) s stands for the ”optimal path” or the ”favorite path” of the polymer,although this path is neither necessarily continuous, nor necessarily unique. Similarly to whatwe have done previously, we define the overlap with the favorite path R ∗ t as the fraction of timeany path B stays next to the favorite path: R ∗ t = R ∗ t ( B, ω ) = 1 t Z t B s ∈ U ( Y ( t ) s ) d s. (5.5.36)As discussed before, a question of interest is the asymptotic behavior, as t → ∞ , of R t andof R ∗ t , and we will see in Theorem 5.5.6 that they are related. In particular, we are interestedin determining the regions were one can prove positivity in the limit of these quantities, whichcan be seen as localization properties of the polymer.Recall the notations D = { ( β, ν ) : ψ ( β, ν ) = 0 } , L = { ( β, ν ) : ψ ( β, ν ) > } , of the high and low temperature regions, which are delimited by the critical lines β − c ( ν ) and β + c ( ν ) (cf. Definition 2.5.1). We saw in theorem 5.5.2 that in the D region, lim t →∞ P [ J t ] = 0.On the other hand, Proposition 5.5.1 tells us that the limit inferior of P [ J t ] is always positivein the region L ′ , where L ′ = (cid:26) β > , ν > (cid:18) ∂ψ∂β (cid:19) + > (cid:27) ∪ (cid:26) β < , ν > (cid:18) ∂ψ∂β (cid:19) − < (cid:27) . (5.5.37)From the preceeding considerations, we know that L ′ ⊂ L , (5.5.38)and a still open question is whether L ′ = L or not. Remark 5.5.3.
It is a direct consequence of the monotonicity of ψ (point 3 of proposition2.4.1), that the inequalities on the derivatives of ψ appearing in (5.5.37) , once replaced by largeinequalities, are always verified: β < ⇒ (cid:18) ∂ψ∂β (cid:19) − ≤ , β > ⇒ (cid:18) ∂ψ∂β (cid:19) + ≥ . We first state some results about the localized region:
Proposition 5.5.4. (i) For any fixed ν > and for large enough positive β , ( β, ν ) ∈ L ′ .(ii) For all ( β, ν ) ∈ L ′ , lim inf t →∞ P [ J t ] > .Proof. To prove (i), we use a result of [19, Th. 2.2.2.(b)] where it is shown that there exists apositive constant C = C ( r, ν, d ), such that, for fixed ν, r and β large enough, p ( β, ν ) ≤ C λ / . By convexity of p in β , we get that for large enough β , (cid:18) ∂p∂β (cid:19) + ≤ p ( β + 1 , ν ) − p ( β, ν ) < νr d e β , so that (cid:16) ∂ψ∂β (cid:17) + is indeed positive when β is big enough.The property (ii) is given by proposition 5.5.1. PPE Remark 5.5.5.
We stress on how strong is the above claim (ii). For ( β, ν ) ∈ L ′ , it impliesthat there exist C > , δ > such that lim inf t →∞ P P β,ωt ⊗ (cid:2) R t ( B, ˜ B ) ≥ δ (cid:3) ≥ C .
In contrast, if β = 0 , there is some C ′ > such that P ⊗ (cid:2) R t ( B, ˜ B ) ≥ δ (cid:3) ≤ e − C ′ t for all large enough t . Observe that the properties of R t and R ∗ t are comparable in the following sense: Theorem 5.5.6.
There exists a constant c = c ( d, r ) in (0 , , such that c (cid:16) P P β,ωt (cid:2) R ∗ t (cid:3)(cid:17) ≤ P [ J t ] ≤ r d P P β,ωt (cid:2) R ∗ t (cid:3) . (5.5.39) In particular, we get that for all β ∈ L ′ , lim inf t →∞ P P β,ωt (cid:2) R ∗ t (cid:3) ≥ r − d lim inf t →∞ P [ J t ] > . (5.5.40) Remark 5.5.7.
Note that equation (5.5.40) gives another feature of path localization of thepolymer in the L ′ region: we can find a ”path” depending only on the environment (here, wefound that Y ( t ) does the job) such that the expected proportion of time the random polymerspends in the neighborhood of that ”path” is bounded away from 0 as t → ∞ . Under the Gibbsmeasure, the random polymer sticks to that particular ”path”. Even though that ”path” is notsmooth – in fact, it has long jumps – it is an interesting object which sumarizes the attractiveeffect of the medium.Proof. To prove the first point, it is enough to show that there is a c ∈ (0 , cP β,ωt (cid:2) R ∗ t (cid:3) ≤ J t ≤ r d P β,ωt (cid:2) R ∗ t (cid:3) , (5.5.41)the proposition being then a simple consequence of Jensen’s inequality. For the right-hand sideinequality of (5.5.41), observe that by Fubini’s theorem, P β,ωt ⊗ P β,ωt (cid:2) R t (cid:3) = 1 t Z t Z P β,ωt [ χ s,x ] d s d x. ≤ t Z t max x ∈ R d P β,ωt ( B s ∈ U ( x )) d s × Z R d P β,ωt (cid:2) χ s,x (cid:3) d x = r d P β,ωt [ R ∗ t ] . In order to obtain the left-hand side inequality, let r d denote the radius of the ball U ( x )and y be any point of R d . By Cauchy-Schwarz’s inequality, Z B( y,r d / P β,ωt (cid:0) B s ∈ U ( z ) (cid:1) d z ! ≤ (cid:12)(cid:12) B( y, r d / (cid:12)(cid:12) Z R d P β,ωt (cid:0) B s ∈ U ( z ) (cid:1) , and since for every z in B( y, r d / y, r d /
2) is included in U ( z ), this inequality leadsto the following: Z R d P β,ωt (cid:0) B s ∈ U ( z ) (cid:1) d z ≥ d r d Z B( y,r d / P β,ωt (cid:0) B s ∈ B( y, r d / (cid:1) d z ! = r d d P β,ωt (cid:0) B s ∈ B( y, r d / (cid:1) . (5.5.42) PPE c ′ = c ′ ( d ) be the minimal number of copies of B( y, r d /
2) necessary to cover U ( y ).Then, by additivity of P β,ωt ,max y ∈ R d P β,ωt (cid:0) B s ∈ U ( y ) (cid:1) ≤ c ′ max y ∈ R d P β,ωt (cid:0) B s ∈ B( y, r d / (cid:1) . (5.5.43)Putting things together and integrating on [0 , t ], we finally get that c t Z t max y ∈ R d P β,ωt (cid:0) B s ∈ U ( y ) (cid:1) d s ≤ t Z t Z P β,ωt [ χ s,x ] d s d x, where c = ( c ′ ) − r d / d , from which the left-hand side inequality of (5.5.41) can be obtained byapplying Jensen’s inequality with probability measure d s/t on [0,t].The second point of the theorem is then a consequence of the first point and proposition5.5.1. Remark 5.5.8.
Formulas like (5.5.39) and (5.5.41) can be called 2-to-1 formulas since theyrelate quenched expectations for two independent polymers to expectations for only one polymer(and involving the optimal path). As we have seen from the computations, stochastic analysisbrings in second moments, involving 2 replicas of the polymer path. Then, using such formulas,the information is reduced to one polymer path interacting with the favorite path.
Formulas for variance and concentration
In this chapter, we introduce the critical exponents of the model and relations between them.The starting points are precise formulas for fluctuations of the partition function (variance andlarge deviations).
There are different ways of defining the critical exponents, see for example [15, 43]. We willnot enter the finest details, and we stay at an intuitive level. Although it is not clear that thesedefinitions are all equivalent, the main idea is that the critical exponents are two reals ξ ⊥ and ξ k such that sup ≤ s ≤ t | B s | ≈ t ξ ⊥ ( d ) and ln Z t − P [ln Z t ] ≈ t ξ k ( d ) as t → ∞ . (6.1.1)The ”wandering exponent” ξ ⊥ is the exponent for the asympotic transversal (or ”perpendic-ular”) fluctuations of the path, with respect to the time axis. The polymer is said to be diffusive when ξ ⊥ = 1 / super-diffusive when ξ ⊥ > /
2. One of the conjectures in polymers is that diffusivity should occure in weak disorder,while super-diffusivity should take place in the strong disorder setting. The number ξ k denotesthe critical exponent for the longitudinal fluctuation of the free energy.The study of these exponents goes beyond the polymer framework. The reason is that theyare expected to take the same value in many different statistical physics models describinggrowth phenomena. In dimension d = 1 this family is called the KPZ universality class (seeSection 10). It is conjectured in the physics literature [38] that the two exponents shoulddepend on one other, in the way that ξ k ( d ) = 2 ξ ⊥ ( d ) − , ∀ d ≥ d = 1, it is conjectured that ξ ⊥ = 2 / ξ k = 1 / β . Fornow, this has only been proven for solvable models of polymers: Sepp¨al¨ainen’s discrete log-gamma polymer [55], O’Connell-Yor semi-discrete polymer [47, 56], and also for the the KPZpolymer [6].In dimension d ≥
2, essentially nothing is known. Let’s simply mention the (rough) bounds0 ≤ ξ k ≤ / , / ≤ ξ ⊥ ≤ / , where the last one will be proved in section 7.A way to approach ξ k is to consider the variance of ln Z t . In what follows, we give a formulato express the variance of ln Z t in terms of a stochastic integral, which is obtained through aClark-Ocone type martingale representation. 34 PPE It is a consequence of Itˆo’s work on iterated stochastic integrals [32] any that square-integrablefunctionals of the Brownian motion can be written as the sum of a constant and an Itˆo inte-gral. In [16], Clark extended this result to a wider range of functionals, and showed that anymartingale that is measurable with respect to the Brownian motion filtration, could be repre-sented as a stochastic integral martingale. Clark was also able to compute the integrand of therepresentation, for a special class of functionals. Ocone then showed [49] that this computationwas linked to Malliavin’s calculus, and generalised this idea to a larger class of functionals.Such representations - called Clark-Ocone representations - also exist in the framework offunctionals of a Poisson processes. Denote by ω s − the restriction of ω on [0 , s ) × R d , andconsider the derivative operator D ( s,x ) F ( ω ) := F ( ω + δ s,x ) − F ( ω ) . (6.2.3)We have: Theorem 6.2.1. [42, theorem 3.1]
Let F = F ( ω ) be a functional of the Poisson process, suchthat P [ F ] < ∞ . Then, P Z P [ D ( s,x ) F ( ω ) | ω s − ] d s d x < ∞ , (6.2.4) and we have for all u ≥ , that P -a.s. P [ F ( ω ) | ω u ] = P [ F ( ω )] + Z [0 ,u ] × R d P [ D ( s,x ) F ( ω ) | ω s − ]¯ ω (d s d x ) . (6.2.5)This proves that the square integrable martingale ( P [ F ( ω ) | ω u ]) u ≥ admits a stochastic in-tegral martingale representation, with predictable integrand P [ D ( s,x ) F ( ω ) | ω s − ]. To lighten the writing, we will denote by G s − the sigma-field generated by ω s − and P G s − willstand for the expectation knowing G s − .Using Jensen’s inequality and Tonelli’s theorem, it is easy to check that ln Z t is a squareintegrable function of ω . Hence, the process (cid:0) P [ln Z t | ω u ] (cid:1) u ∈ [0 ,t ] is a martingale which admits aClark-Ocone type representation: P [ln Z t | ω u ] = P [ln Z t ] + Z [0 ,u ] × R d P G s − (cid:2) D ( s,x ) ln Z t (cid:3) ¯ ω (d s d x ) , (6.3.6)where D ( s,x ) F ( ω ) = ln P (cid:2) e βω ( V t ) e βχ s,x (cid:3) Z t = ln (cid:16) λP β,ωt [ χ s,x ] (cid:17) . As a consequence, one can express the variance of ln Z t via (6.3.6), using the formula for thevariance of a Poisson integral (5.1.5), and find thatVar(ln Z t ) = P Z [0 ,t ] × R d P G s − h ln (cid:16) λP β,ωt [ χ s,x ] (cid:17)i ν d s d x. (6.3.7)This variance formula leads us to the following theorem: PPE Theorem 6.3.1. (i) The following lower and upper bounds on the variance hold:
Var(ln Z t ) ≥ c − P Z [0 ,t ] × R d P G s − (cid:2) P β,ωt [ χ s,x ] (cid:3) ν d s d x, (6.3.8)Var(ln Z t ) ≤ c P Z [0 ,t ] × R d P G s − (cid:2) P β,ωt [ χ s,x ] (cid:3) ν d s d x, (6.3.9) where c − = 1 − e −| β | and c + = e | β | − .In particular, Var(ln Z t ) ≤ c tν P [ J t ] . (6.3.10) (ii) Letting c = νc exp( c + ) , the following concentration estimate holds: P (cid:0)(cid:12)(cid:12) ln Z t − P [ln Z t ] (cid:12)(cid:12) > u (cid:1) ≤ (cid:18) −
12 ( u ∧ u ct ) (cid:19) . (6.3.11) Remark 6.3.2.
Recalling Theorem 5.5.2, the inequality (6.3.10) suggests that the varianceshould be smaller in weak disorder than in strong disorder. It also shows that for all d ≥ , wehave ξ k ( d ) ≤ / .Proof. The two first bounds on the variance are a consequence of the fact that, for all u ∈ [0 , c − u ≤ | ln(1 + λu ) | ≤ c + u. (6.3.12)Then, apply Jensen’s inequality to the conditional expectation in the right-hand side of (6.3.9)and use Fubini’s theorem such thatVar(ln Z t ) ≤ c P Z [0 ,t ] × R d P G s − (cid:2) P β,ωt [ χ s,x ] (cid:3) ν d s d x ≤ c P Z [0 ,t ] × R d P β,ωt [ χ s,x ] ν d s d x = c tν P [ J t ] , by definition of J t . This completes the proof of (i). To prove (6.3.11), we first denote by Y t,u the mean-zero martingale part appearing in (6.3.6), i.e. Y t,u := Z [0 ,u ] × R d P G s − ln (cid:16) λP β,ωt [ χ s,x ] (cid:17) ¯ ω (d s d x ) . Then, letting ϕ ( v ) = e v − v − a ∈ [ − , M t,u ) u ∈ [0 ,t ] as the exponentialmartingale associated to ( Y t,u ) u ∈ [0 ,t ] : M t,u = exp aY t,u − Z [0 ,u ] × R d ϕ (cid:16) a · P G s − ln (cid:0) λP β,ωt [ χ s,x ] (cid:1)(cid:17) ν d s d x ! . By (6.3.12) and the observations that χ is less than 1 and that | ϕ ( v ) | ≤ e | v | v / v , wehave for a ∈ [ − , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z [0 ,t ] × R d ϕ (cid:16) a · P G s − ln (cid:16) λP β,ωt [ χ s,x ] (cid:17)(cid:17) ν d s d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ e c + c a Z [0 ,t ] × R d P G s − (cid:2) P β,ωt [ χ s,x ] (cid:3) ν d s d x ≤ c a Z [0 ,t ] × R d P G s − (cid:2) P β,ωt [ χ s,x ] (cid:3) d s d x = c a t, PPE c = νc e c + and where the second inequality was obtained using Jensen’s inequality.If one denotes by b t,u the integral term in the definition of M t,u , we just showed that b t,t ≤ ca t/
2, so by Markov’s inequality and the martingale property, we obtain that P (cid:0) ln Z t − P [ Z t ] > u (cid:1) = P (cid:0) M t,t > exp( au − b t,t ) (cid:1) ≤ exp( ca t/ − au ) . (6.3.13)This implies (6.3.11) after minimizing the bound for a ∈ [ − , Corollary 6.3.3.
For all ε > and as t → ∞ , ln Z t − P [ln Z t ] = O (cid:16) t ε (cid:17) , P -a.s. (6.3.14) Proof.
Equation (6.3.11) implies that for large enough k ∈ N , P h | ln Z k − P ln Z k | > k ε i ≤ (cid:0) − t ε c (cid:1) , (6.3.15)which is summable. By Borel-Cantelli lemma, we obtain that, P -almost surely, | ln Z k − P [ln Z k ] | ≤ k ε for k large enough . To extend this to any t ≥ Lemma 6.3.4.
Let h > . For all ≤ s ≤ h , − c + δ t ( h ) ≤ ln Z t + s − ln Z t ≤ c + δ t ( h ) , (6.3.16) where δ t ( h ) = Z [ t,t + h ] × R d P β,ωs − [ χ s,x ] ω (d s d x ) , is such that for any ε > , δ t ( h ) = O (cid:0) t ε (cid:1) , P -a.s. (6.3.17) Proof.
We get from the integral writing of ln Z t (5.2.6) thatln Z t + s − ln Z t = Z [ t,t + h ] × R d ω (d s d x ) ln (cid:0) λP β,ωs − [ χ s,x ] (cid:1) . Hence, (6.3.16) is simply obtained with (6.3.12).Now, introduce the martingale M t = Z t ω (d s d x ) P β,ωs − [ χ s,x ] − νr d t, which has bracket h M i t = ν R t I s d s ≤ νr d t . Note that δ t ( h ) = M t + h − M t + h , so that (6.3.17)is thus a consequence of the martingale properties (5.4.21) and (5.4.22), since in the case where h M i ∞ is infinite, then | M t | = o (cid:0) h M i ε t (cid:1) as t → ∞ .In order to illustrate the general strategy, we now mention a consequence of theorem 6.3.1(i). PPE Corollary 6.3.5.
Let β = 0 , ξ and C > . There exists a constant c = c ( d, C ) ∈ (0 , ∞ ) ,such that lim inf t →∞ t − (1 − dξ ) Var(ln Z t ) ≥ c lim inf t →∞ inf ≤ s ≤ t (cid:16) P P β,ωt (cid:16) | B s | ≤ C + Ct ξ (cid:17)(cid:17) ≥ c lim inf t →∞ (cid:18) P P β,ωt (cid:18) sup ≤ s ≤ t | B s | ≤ C + Ct ξ (cid:19)(cid:19) . The result suggests that χ ( d ) ≥ − dξ ( d )2 . For more details on the result and the proof, we refer the reader to Corollary 2.4.3 in [19].
Cameron-Martin transform and applications
In this section we extensively use the property that the a-priori measure for the polymer pathis Wiener measure. A tilt on the polymer path reflects into a shift on the environment.
We extend the shear transformation (4.1.3 – 4.1.4) to non-linear shifts ϕ : R + → R d by definingˆ τ ϕ f : s f ( s ) + ϕ ( s ) , ˆ τ ϕ ◦ (cid:0) X i δ ( t i ,x i ) (cid:1) = X i δ ( t i ,x i + ϕ ( t i )) , (7.1.1)so that ˆ τ ϕ = τ ξ when ϕ ( t ) = tξ and that ˆ τ ϕ ◦ ω has same law as ω .Let ϕ ∈ H , loc := (cid:8) ϕ ∈ C ( R + , R d ); ϕ (0) = 0 , ˙ ϕ ∈ L (cid:9) where the dot denotes time deriva-tive. Introduce the probability measure on the path space P ϕ which restriction to F t hasdensity relative to P given by (cid:0) dP ϕ dP (cid:1) |F t = exp (cid:26)Z t ˙ ϕ ( s ) dB ( s ) − Z t | ˙ ϕ ( s ) | ds (cid:27) for all t >
0, with B the canonical process. Then, by Cameron-Martin theorem, under themeasure P ϕ the process W ( t ) = B ( t ) − ϕ ( t ) is a standard Brownian motion, and as in (4.2.7)we write P h exp { βω ( V t ( B )) } e R t ˙ ϕ ( s ) dB ( s ) − R t | ˙ ϕ ( s ) | ds i = P ϕ [exp { βω ( V t (ˆ τ ϕ ( W )) } ]= P [exp { βω ( V t (ˆ τ ϕ ( B ))) } ]= P [exp { βω (ˆ τ ϕ ( V t ( B ))) } ]= P [exp { β (ˆ τ − ϕ ◦ ω )( V t ( B )) } ]= Z t (ˆ τ − ϕ ◦ ω, β ) , yielding P β,ωt (cid:20) exp { Z t ˙ ϕ ( s ) dB ( s ) } (cid:21) = exp { Z t | ˙ ϕ ( s ) | ds } × W t (ˆ τ − ϕ ◦ ω, β ) W t ( ω, β ) . (7.1.2) In this section we assume that β ∈ ( ¯ β − c , ¯ β + c ), more precisely that W ∞ ( ω, β ) = lim t W t ( ω, β ) > ϕ ∈ H (i.e., ˙ ϕ ∈ L ), we derive from (7.1.2) that P β,ωt (cid:20) exp { Z t ˙ ϕ ( s ) dB ( s ) } (cid:21) −→ exp { k ˙ ϕ k } × W ∞ (ˆ τ − ϕ ◦ ω, β ) W ∞ ( ω, β ) (7.2.3)a.s. as t → ∞ .In view of (7.1.2), a natural question is continuity of W ∞ in the ω variable: how does thelimit depend on the environment ? 39 PPE Lemma 7.2.1.
Assume that W ∞ ( ω, β ) > . Let ϕ T ∈ C ( R + , R d ) be a family indexed by T > with ϕ T (0) = 0 which vanishes locally uniformly, i.e., ∀ t > , k ϕ T k ∞ ,t := sup {| ϕ T ( s ) | ; s ∈ [0 , t ] } → T → ∞ . Then we have, as T → ∞ , W ∞ (ˆ τ ϕ T ◦ ω, β ) −→ W ∞ ( ω, β ) in L − norm , (7.2.4) W T (ˆ τ ϕ T ◦ ω, β ) −→ W ∞ ( ω, β ) in L − norm . (7.2.5)The result can be compared to lemma 3.2.1, where we have already considered the effect ofa shift on the environment. In the notation of (7.1.1), that lemma deals with constant shifts ϕ = ϕ ( x ) such that ϕ ( x ) ( t ) ≡ x for all t , and implies that x W ∞ (ˆ τ ϕ ( x ) ◦ ω ) is Lipschitz continuous from R d to L . In the above lemma 7.2.1, the shift is not anymore constant.
Proof.
Fix t > W ∞ (ˆ τ ϕ T ◦ ω, β ) − W ∞ ( ω, β ) as { W ∞ (ˆ τ ϕ T ◦ ω, β ) − W t (ˆ τ ϕ T ◦ ω, β ) } + { W t (ˆ τ ϕ T ◦ ω, β ) − W t ( ω, β ) } + { W t ( ω, β ) − W ∞ ( ω, β ) } . Now, using triangular inequality and invariance in law of ω under the shear transformation, weget k W ∞ (ˆ τ ϕ T ◦ ω, β ) − W ∞ ( ω, β ) k ≤ k W ∞ ( ω, β ) − W t ( ω, β ) k + k W t (ˆ τ ϕ T ◦ ω, β ) − W t ( ω, β ) k =: 2 ε ( t ) + ε t ( ϕ T ) (7.2.6)with ε ( t ) = k W ∞ − W t k ∞ and ε t ( · ) defined by the above formula. By assumption on β andproposition 3.2.2, we have lim t →∞ ε ( t ) = 0. On the other hand, for fixed t , W t (ˆ τ ϕ T ◦ ω, β ) → W t ( ω, β ) a.s. as T → ∞ , and the variables ( W t (ˆ τ ϕ T ◦ ω, β ); T > W t ( ω, β ) are uniformlyintegrable. Thus, the above convergence holds in L , which, combined with (7.2.6), completesthe proof of (7.2.4).The proof of (7.2.5) is quite similar, writing this time the difference W T (ˆ τ ϕ T ◦ ω, β ) − W ∞ ( ω, β ) = { W T (ˆ τ ϕ T ◦ ω, β ) − W ∞ (ˆ τ ϕ T ◦ ω, β ) } + { W ∞ (ˆ τ ϕ T ◦ ω, β ) − W ∞ ( ω, β ) } , observing that the first term in the right-hand side has L -norm equal to k W T − W ∞ k , thatthe second one vanishes a.s. and is uniformly integrable. This completes the proof.For a ∈ R d , t > ϕ a,t ( s ) = s ∧ t √ t a . From (7.1.2) with ϕ = ϕ a,t , we have P β,ωt (cid:20) e a · B ( t ) √ t (cid:21) = e | a | / × W t (ˆ τ − ϕ a,t ◦ ω, β ) W t ( ω, β ) . (7.2.7) Theorem 7.2.2.
Assume weak disorder, i.e., that W ∞ > . Then, as t → ∞ , P β,ωt (cid:20) e a · B ( t ) √ t (cid:21) P −→ e | a | / . PPE Proof.
Note that the family ( − ϕ a,t , t >
0) satisfies the assumptions of lemma 7.2.1. Writingformula (7.2.7) as P β,ωt (cid:20) e a · B ( t ) √ t (cid:21) − e | a | / = e | a | / × W t (ˆ τ − ϕ a,t ◦ ω, β ) − W t ( ω, β ) W t ( ω, β ) , we see from Slutsky’s lemma that this quantity vanishes as t → ∞ . Remark 7.2.3.
In a suitable sense, this result shows that the polymer is diffusive if W ∞ > .Its interest is that it covers the full weak disorder region, in contrast to [18, Th. 2.1.1] whichonly applies to the L region. It can be viewed as a step to prove diffusivity at weak disorderregion. In this section, β ∈ R is arbitrary. Theorem 7.3.1.
For P -a.e. realization of the environment ω , the following holds.(i) For all Borel A ⊂ R d , − inf { | ξ | ξ ∈ ˚ A } ≤ lim inf t →∞ t ln P β,ωt h B ( t ) t ∈ A i ≤ lim sup t →∞ t ln P β,ωt h B ( t ) t ∈ A i ≤ − inf { | ξ | ξ ∈ A } . (ii) Let t n be a positive sequence increasing to ∞ , let χ > such that X n ≥ P ( | ln Z t n ( ω, β ) − P [ln Z t n ( ω, β )] | > t χn ) < ∞ , (7.3.8) and let ξ > (1 + χ ) / . Then, for all Borel A ⊂ R d , − inf { | a | a ∈ ˚ A } ≤ lim inf t →∞ t − (2 ξ − n ln P β,ωt n h B ( t n ) t ξn ∈ A i ≤ lim sup t →∞ t − (2 ξ − n ln P β,ωt n h B ( t n ) t ξn ∈ A i ≤ − inf { | a | a ∈ A } . Part (i) is the almost sure large deviation principle for the polymer endpoint. The ratefunction a
7→ | a | / χ > ξ k , we derive that the polymer endpointlies at time t n within a distance t ξ k + εn with overwhelming probability for all positive ε . Hencewe get a relation between characteristic exponents ξ ⊥ ≤ ξ k . (7.3.9)Since ξ k ≤ / d : ξ ⊥ ≤ / . (7.3.10) PPE Proof.
We start with (i). Using (7.1.2) with ϕ = ϕ t given by ϕ t ( s ) = ( s ∧ t ) a , we getln P β,ωt [ e a · B ( t ) ] = t | a | W t ( τ a ◦ ω, β ) − ln W t ( ω, β )= t | a | (cid:0) ln W t ( τ a ◦ ω, β ) − P [ln W t ( τ a ◦ ω, β )] (cid:1) + (cid:0) ln W t ( ω, β ) − P [ln W t ( ω, β )] (cid:1) , and by theorem 2.3.1 the set Ω a of environments such thatlim t →∞ t ln P β,ωt [ e a · B ( t ) ] = | a | P -measure. We now claim that on the event Ω = T a ∈ Q d Ω a the above limit holds forall a ∈ R d . Indeed, the maps a t ln P β,ωt [ e a · B ( t ) ] are convex, and the convergence is locallyuniform on the closure R d of Q d , see Th. 10.8 in [52]. Then, the large deviation principle (i)follows from the G¨artner-Ellis-Baldi theorem ([23], p.44, Th.2.3.6), with the rate function | ξ | / {| a | / a ∈ R d } . For the proof of (ii) we proceed as above. By (7.1.2) with ϕ t ( s ) = ( s ∧ t ) t ξ − a , we firstwrite t − (2 ξ − ln P β,ωt [ e t ξ − a · B ( t ) ] = | a | t − (2 ξ − (cid:0) ln W t (ˆ τ − ϕ t ◦ ω, β ) − P [ln W t ( ω, β )] (cid:1) + t − (2 ξ − (cid:0) ln W t ( ω, β ) − P [ln W t ( ω, β )] (cid:1) . Now, in order to take the limit we restrict to the sequence t = t n : using Borel-Cantelli lemma,we indeed get the a.s. limit (7.3.11) along the sequence t n . From then on, the other argumentsgo through without any change. Phase diagram in the ( β, ν ) -plane, d ≥ In this section, r > β, ν . Recall ψ form (2.4.15) and ψ t ( β, ν ) = νλ ( β ) r d − t − P [ln Z t ( ω, β )] from (2.4.18).Recall the notations D = { ( β, ν ) : ψ ( β, ν ) = 0 } , L = { ( β, ν ) : ψ ( β, ν ) > } , Crit = D \ L . D is is the delocalized phase, L the localized phase.They are also high temperature/low densityand low temperature/high density phases respectively. They are separated by the critical curve Crit = { ( β + c ( ν ) , ν ); ν > } ∪ { ( β − c ( ν ) , ν ); ν > ν c } . We have seen that, in dimension d = 1 or 2, D reduces to the axis β = 0, so we assume d ≥ ν c = sup { ν > β − c = −∞} . (8.0.1)Then, ν c ∈ (0 , ∞ ), and ν > ν c ⇐⇒ β − c > −∞ . A central question in polymer models is to estimate the critical curve [24, 27].
We follow the idea of [20], that is tofind curves ν ( β ) along which ψ is monotone.We now sketch the strategy. By computing the derivative with the chain rule ddβ ψ t ( β, ν ( β )) = ν ′ ∂∂ν ψ t + ∂∂β ψ t , one sees that, along the smooth curve C αa ν ( β ) = a | λ ( β ) | − α (8.1.2)for positive constants a, α , it takes the amenable form t ∂∂β ψ t ( β, ν ( β )) = ν ′ × P "Z (0 ,t ] × R d h α ( λP β,ωt [ χ s,x ]) dsdx , (8.1.3)where h α ( u ) = u − u α (1 + u ) − ln(1 + u ) (8.1.4)for u > −
1. Now, the questions boils down to controling the sign of the function on the relevantinterval with endpoints 0 and λ ( β ): h α ( u ) ≥ α = 2 and u ≥ , ≤ α = 2 and u ∈ ( − , , ≤ β > , α ≤ α ( β ) and u ∈ [0 , λ ] , ≥ β < , α ≥ α ( β ) and u ∈ [ λ, , with α ( β ) = ( e β − e β ( e β − − β ) , α (0) = 2. With this at hand, we can bound the critical curve fromabove and below with curves of the form (8.1.2) and specific α ’s, as indicated on Figure 8.1.43 PPE C αa denotes the curve (8.1.2). We start with an estimate of the free energy. In [20, Th. 5.3.1], the asymptotics of the freeenergy is determined as νβ diverges and β remains bounded. We state it below, it is key forour results. Lemma 8.2.1.
Let β ∈ (0 , ∞ ) arbitrary. Then, as νβ → ∞ and | β | ≤ β , we have p ( ν, β ) = βνr d + O (( νβ r d ) / . We refer to the above paper for the involved, technical proof.Among all the results, we mention:
Theorem 8.2.2.
Let d ≥ .(i) the functions β ± c ( ν ) are locally Lipshitz and strictly monotone;(ii) we have β + c ( d, ν ) ≍ ln(1 /ν ) as ν ց
0; (8.2.5) (ii) we have | β ± c ( d, ν ) | ≍ / √ ν as ν ր ∞ . (8.2.6) The derivative of ψ t in β has been obtained in (2.4.17). We explain how to obtain the derivativein ν , and for clarity, in the sequel of the section we write P = P ν to make the dependence in ν explicit. For positive functions we write f ( x ) ≍ g ( x ) as x → x if the ratio remains bounded from 0 and from ∞ as x → x . PPE Lemma 8.3.1.
We have ∂∂ν P ν [ln Z t ( β, ω )] = Z [0 ,t ] × R d dsdx P ν ln[1 + ln P β,ωt ( χ s,x )] . (8.3.7) Proof. : For k ≥
1, let Z t,k = P [ e βω ( V t ) ; A k ] , with A k = { B s ∈ [ − k, k ] d , ∀ s ≤ t } , and p t,k ( β, ν ) = t − P ν ln Z t,k . Let K r = [ − k − r, k + r ] d and K t = (0 , t ] × K r . By Proposition3.1.4, P ν [ln Z t,k ] = P [ ρ t,ν ln Z t,k ] with ρ t,ν = exp ( ω t ( K t ) ln ν − ( ν − t | K r | ) . Thus, tp t,k ( β, ν ) is differentiable in ν , with derivative1 ν P ν (cid:20)Z K t ω ( dsdx ) ln Z t,k (cid:21) (2.4.16) = Z K t dsdx P ν ln Z t,k ( ω + δ s,x ) Z t,k ( ω )= Z [0 ,t ] × R d dsdx P ν ln (cid:16) P β,ωt [ χ s,x | A k ] (cid:17) . Now, we write tp t,k ( β, ν ) − tp t,k ( β,
1) = Z ν dν ′ Z [0 ,t ] × R d dsdx P ν ′ ln (cid:16) P β,ωt [ χ s,x | A k ] (cid:17) . By dominated convergence theorem (see details in [20], Lemma 7.2.1), we can take the limit k → ∞ , and obtain the desired statement.We come to the core of the proof. With the derivatives of ψ t in both variables, from Lemma8.3.1 and (2.4.17), we obtain t ddβ ψ t ( β, ν ( β )) = ν ′ t ∂∂ν ψ t + t ∂∂β ψ t = ν ′ × P Z [0 ,t ] × R d dsdx ( λP β,ωt [ χ s,x ] + νν ′ e β λ P β,ωt [ χ s,x ] λP β,ωt [ χ s,x ] − ln (cid:0) λP β,ωt [ χ s,x ] (cid:1)) . Recall that e β = λ ′ ; we recover the simpler formula (8.1.3) along the curves of equation λ ′ νλν ′ = − α , that is, the curves h α from (8.1.4). From then on, the rest of the proof is a tedious butelementary exercise in calculus, performed in [20]. We will not dive any further in the detailsof the proof, that the reader can find in this reference together with many fine estimates. Wesummarize the section by giving a qualitative picture of the phase diagram in Figure 8.2. PPE d ≥ Complete localization
As we send some parameters to 0 or ∞ , the present model converges to other related polymermodels. A first instance is the intermediate disorder regime of section 10, where parameters β, ν, r are scaled with the polymer size, see (10.4.29). Another instance is the mean field limit: independently of the polymer length, we let ν → ∞ and β → νβ → b ∈ (0 , ∞ ).Then, the rewards given by the Poisson medium get denser and weaker in this asymptotics sothat they turn into a Gaussian environment, given by generalized Gaussian process g ( t, x ) withmean 0 and covariance E [ g ( t, x ) g ( s, y )] = b δ ( t − s ) | U ( x ) ∩ U ( y ) | , where | · | above denotes the Lebesgue measure. In other words, the environment is gaussian,which is correlated in space but not in time – it is Brownian-like. Here, the limit of our modelis a Brownian directed polymer in a Gaussian environment, introduced in [54], with partitionfunction Z t = P (cid:20) exp { Z t g ( s, B ( s )) ds } (cid:21) . We do not elaborate this asymptotics, but instead we focus on the case b = ∞ . This corresponds to letting, independently of the polymer length, ν → ∞ , | β | ≤ β , such that νβ → ∞ . (9.2.1)(The parameter r is kept fixed.) Precisely, we first let t → ∞ and then take the limit (9.2.1). Theorem 9.2.1.
Under the assumption (9.2.1) , − O (cid:16) ( νβ ) − / (cid:17) ≤ lim inf t →∞ P (cid:2) P β,ωt ( R ∗ t ) (cid:3) ≤ lim sup t →∞ P (cid:2) P β,ωt ( R ∗ t ) (cid:3) ≤ − O (cid:16) ( νβ ) − / (cid:17) . This statement describes the strong localization properties of the polymer path. The time-average t R t B s ∈ U ( Y ( t ) ( s )) ds is the time fraction the polymer spends together with the favouritepath. We know that when ( ∂ψ ) / ( ∂β ) = 0 the time fraction is positive. The claim here is thatit is almost the maximal value 1, in the limit (9.2.1). For a benchmark, we recall that, for thefree measure P , for all smooth path Y and all δ >
0, there exists a positive C such that forlarge t , P (cid:18) t Z t B s ∈ U ( Y ( s )) ds ≥ δ (cid:19) ≤ e − Ct (9.2.2)(In fact, it is not difficult to see (9.2.2) for Y ≡ Y .)47 PPE Proof.
Let ˆ φ t ( β ) = t − ( P ln Z t − tνβr d ). We assume β ≥
0, the other case being similar. Byconvexity of ˆ φ t ,ˆ φ t (2 β ) − ˆ φ t ( β ) convex ≥ β ∂ ˆ φ t ∂β ( β ) (2.4.17) = βνλ ( β ) t Z (0 ,t ] × R d dsdx P P β,ωt [ χ s,x ] − P β,ωt [ χ s,x ] λ ( β ) P β,ωt [ χ s,x ] . Bounding from above the denominator in the integral by e β , we get1 − P P β,ωt ⊗ [ R t ] ≤ e β ˆ φ t (2 β ) − ˆ φ t ( β ) βλνr d . (9.2.3)Now, using the bound in Lemma 8.2.1, we deriveR . H . S . (9.2.3) = O (( νβ ) − / . Similarly, ˆ φ t ( β ) − ˆ φ t ( β/ convex ≤ ( β/ ∂ ˆ φ t ∂β ( β )= βνλ ( β )2 t Z (0 ,t ] × R d dsdx P P β,ωt [ χ s,x ] − P β,ωt [ χ s,x ] λ ( β ) P β,ωt [ χ s,x ] , leading to 1 − P P β,ωt ⊗ [ R t ] ≥ ˆ φ t ( β ) − ˆ φ t ( β/ βλνr d = O (( νβ ) − / , and to the desired result.We can extract fine additional information and geometric properties of the Gibbs measure.For δ ∈ (0 , /
2) define the ( δ, t )-negligible set as N ηδ,t = n ( s, x ) ∈ [0 , t ] × R d : P β,ωt ( χ s,x ) ≤ δ o , and the ( δ, t )-predominant set as P ηδ,t = n ( s, x ) ∈ [0 , t ] × R d : P β,ωt ( χ s,x ) ≥ − δ o . As suggested by the names, N ηδ,t is the set of space-time locations the polymer wants to stayaway from, and P ηδ,t is the set of locations the polymer likes to visit. Both sets depend on theenvironment. Corollary 9.2.2.
For all < δ < / , we have, under the assumption (9.2.1) , lim sup t →∞ P (cid:20) t (cid:12)(cid:12)(cid:12) ( N ηδ,t ∪ P ηδ,t ) ∁ (cid:12)(cid:12)(cid:12)(cid:21) = O (cid:16) ( νβ ) − / (cid:17) , (9.2.4)lim sup t →∞ P P β,ωt (cid:20) t (cid:12)(cid:12)(cid:12) V t ( B ) \ N ηδ,t (cid:12)(cid:12)(cid:12)(cid:21) = O (cid:16) ( νβ ) − / (cid:17) , (9.2.5)lim sup t →∞ P P β,ωt (cid:20) t (cid:12)(cid:12)(cid:12) V t ( B ) ∁ \ P ηδ,t (cid:12)(cid:12)(cid:12)(cid:21) = O (cid:16) ( νβ ) − / (cid:17) . (9.2.6) PPE | · | denotes the Lebesgue measure on R + × R d , and note that |N ηδ,t | = | V t ( B ) ∁ | = ∞ . The limits (9.2.4), (9.2.5), (9.2.6), bring information on how is the corridor around thefavourite path where the measure concentrates for large νβ . We depict the main features forlarge νβ : • most (in Lebesgue measure) time-space locations become negligible or predominant, • most (in Lebesgue and Gibbs measures) negligible locations are outside the tube aroundthe polymer path, • most (in Lebesgue and Gibbs measures) predominant locations are inside the tube aroundthe polymer path.The trace (cid:8) x ∈ R d : P β,ωt ( χ s,x ) ≥ − δ (cid:9) at time t of the ( δ, t )-predominant set is reminiscentof the ǫ -atoms discovered in [64], with ǫ = 1 − δ , and discussed in [7]. These references studythe time and space discrete setting, and restrict to the end point of the polymer. Proof.
It suffices to prove, for all δ ∈ (0 , / tr d (cid:12)(cid:12)(cid:12)n ( s,x ) ∈ [0 , t ] × R d : P β,ωt ( χ s,x ) ∈ [ δ, − δ ] o(cid:12)(cid:12)(cid:12) ≤ δ (1 − δ ) P β,ωt ⊗ (cid:16) − R t (cid:17) (9.2.7) P β,ωt (cid:20) tr d (cid:12)(cid:12)(cid:12) V t ( B ) \ n ( s, x ) : P β,ωt ( χ s,x ) ≤ δ o(cid:12)(cid:12)(cid:12)(cid:21) ≤ − δ P β,ωt ⊗ (cid:16) − R t (cid:17) (9.2.8) P β,ωt (cid:20) tr d (cid:12)(cid:12)(cid:12) V t ( B ) ∁ \ n ( s, x ) : P β,ωt ( χ s,x ) ≥ − δ o(cid:12)(cid:12)(cid:12)(cid:21) ≤ − δ P β,ωt ⊗ (cid:16) − R t (cid:17) . (9.2.9)Note that u (1 − u ) ≥ (1 − δ ) u u<δ + δ (1 − δ ) u ∈ [ δ, − δ ] + (1 − δ )(1 − u ) u> − δ . Setting A s = { x : P β,ωt ( χ s,x ) ∈ [ δ, − δ ] } and writing P β,ωt ⊗ (1 − R t ) = 1 tr d Z t Z R d h P β,ωt ( χ s,x ) − P β,ωt ( χ s,x ) i dxds ≥ tr d Z t Z A s h P β,ωt ( χ s,x ) − P β,ωt ( χ s,x ) i dxds ≥ δ (1 − δ ) 1 tr d Z t (cid:12)(cid:12)(cid:12) { x : P β,ωt ( χ s,x ) ∈ [ δ, − δ ] } (cid:12)(cid:12)(cid:12) ds, which yields (9.2.7). For the next one, we write P β,ωt ⊗ (1 − R t ) = 1 tr d Z t Z R d h P β,ωt ( χ s,x ) − P β,ωt ( χ s,x ) i dxds ≥ (1 − δ ) 1 tr d Z t Z R d P β,ωt ( χ s,x ) P β,ωt χ s,x ) <δ dsdx = (1 − δ ) P β,ωt (cid:20) tr d Z t Z R d P β,ωt ( χ s,x ) <δ,B s ∈ U ( x ) (cid:21) dsdx, which is (9.2.8). The last claim can be proved similarly. d = 1 ) In this chapter, we focus on dimension d = 1, where the polymer is in the strong disorder phaseas soon as β = 0 is kept fixed (Remark 3.4.3). Although it is believed that the model satisfies the following non-standard critical exponents:sup ≤ s ≤ t | B s | ≈ t / and ln Z t − P [ln Z t ] ≈ t / as t → ∞ , (10.1.1)proofs are missing at this moment. It is also expected that the fluctuations of the free energyaround its mean are of Tracy-Widom type: Conjecture 10.1.1.
For all non-zero β, ν and r , there exists some constant σ ( β, ν ) such that,as t → ∞ , ln Z t − p ( β, ν ) tσ ( β, ν ) t / −→ F GOE (10.1.2) where the F GOE is the Tracy-Widom GOE distribution [62] . These properties are characteristics of the KPZ universality class. They are in sharp contrastto the weak disorder regime, where one knows to a large extent that B t ≈ t / (Theorem 7.2.2),and where the free energy ln Z t has order one fluctuations around its mean (3.1.3), which arefeatures of the Edward-Wilkinson universality class.The KPZ universality class is a family of models of random surfaces dynamics that sharenon-gaussian statistics, non-standard critical exponents and scaling relations (3-2-1 in time,space and fluctuations, as in (10.1.1)). Members of this class include some interacting par-ticles systems (asymmetric simple exclusion prosses (ASEP), interacting Brownian motions),paths in random environment (directed polymers, first and last passage percolation), stochas-tic PDEs (KPZ equation, stochastic Burgers equation, stochastic reaction-diffusion equations).The reader may refer to [21] for a non-technical review on the KPZ universality class.The Kardar-Parisi-Zhang (KPZ) equation is the non-linear stochastic partial differentialequation: ∂ H ∂T ( T, X ) = 12 ∂ H ∂X ( T, X ) + 12 (cid:18) ∂ H ∂X ( T, X ) (cid:19) + βη ( T, X ) , (10.1.3)where β ∈ R and η is a random measure on [0 , × R called the space-time Gaussian whitenoise , which verifies that:(i) For all measurable sets A , . . . , A k of [0 , × R , (cid:0) η ( A ) , . . . , η ( A k ) (cid:1) is a centered Gaussianvector.(ii) For all measurable sets A, B of [0 , × R , then P [ η ( A ) η ( B )] = | A ∩ B | . The KPZ equation models the behavior of a random interface growth and was introducedby Kardar, Parisi and Zhang [36] in 1986. It is difficult to make sense of this equation andBertini-Cancrini [11] argued that a possible definition of H β could be given by the so-called Hopf-Cole transformation : H β ( T, X ) = ln Z β ( T, X ) , (10.1.4)50 PPE Z β is the solution of the stochastic heat equation (SHE): ∂ Z β ∂T ( T, X ) = 12 ∂ Z β ∂X ( T, X ) + β Z β ( T, X ) η ( T, X ) . (10.1.5)In a breakthrough paper [3], Amir, Corwin and Quastel were able to describe the pointwisedistribution of H β ( T, X ) by exploiting the weak universality of the ASEP model. It resultsfrom this that the KPZ equation lies in the KPZ universality class.The weak KPZ universality conjecture states that the KPZ equation is a universal objectof the KPZ class. As a general idea, the KPZ equation should appear as a scaling limit atcritical parameters for models that feature a phase transition between the Edward-Wilkinsonclass (4-2-1 scaling) and the KPZ class. This was first verified for the model of ASEP [12],and more recently for the discrete and Brownian directed polymers [2, 22]. The proofs relyon the Hopf-Cole transformation, which enables one to switch between the KPZ equation andthe stochastic heat equation. In this chapter, we essentially summarize the arguments of [22]to explain why the Brownian polymer in Poisson environment model verifies the weak KPZuniversality.
A special case of interest for the SHE, where Z β ( T, X ) can be seen as the point-to-pointpartition function of a directed polymer, placed at X = 0 at time T = 0, is when Z β (0 , X ) = δ ( X ) . (10.2.6)In this case, Z β ( T, X ) can be expressed through the following shortcut (cf. Section 10.3.1): Z β ( T, X ) = ρ ( T, X ) P T,X , (cid:20) : exp : (cid:18) β Z T η ( u, B u )d u (cid:19)(cid:21) , (10.2.7)where ρ ( t, x ) = e − x / t / √ πt .This equation is similar to the definition of the point-to-point partition function a polymerwith Brownian path and white noise environment. Alberts, Khanin and Quastel [1] were in factable to construct a polymer measure with P2P partition function given by Z β ( T, X ). As boththe environment and the paths of the polymer are continuous, it was named the continuumdirected random polymer .Similarly to the Poisson polymer, the P2P free energy F β ( T, X ) can be defined as F β ( T, X ) = ln Z β ( T, X ) ρ ( T, X ) , (10.2.8)so that the free energy of the polymer and the solution of the KPZ equation follow the relation: F β ( T, X ) = H β ( T, X ) + X / T + ln √ πT . (10.2.9) Introduce the renormalized point-to-point partition function: W ( t, x ; ω, β, r ) = ρ ( t, x ) P t,x , h exp { βω ( V t ) − λ ( β ) νr d t } i . (10.2.10) PPE W ( t, x ; ω, β, r ) = W ( t, x ) when no confusion can arise.Compared to Z t ( ω, β ; x ) of (4.1.1), a major difference is that it encorporates the Gaussiankernel as a factor. In the next theorem, we state that the renormalized P2P partition functionverifies a weak formulation of the following stochastic heat equation with multiplicative Poissonnoise : ∂ t W ( t, x ) = 12 ∆ W ( t, x ) + λW ( t − , x )¯ ω (d t × U ( x )) . (10.2.11)When β = 0, it reduces to the usual heat equation. Theorem 10.2.1 (Weak solution) . For all ϕ ∈ D ( R ) and t ≥ , we have P -almost surely Z R W ( t, x ) ϕ ( x )d x = ϕ (0) + 12 Z t d s Z R W ( s, x )∆ ϕ ( x )d x + λ Z R d xϕ ( x ) Z (0 ,t ] × R ¯ ω (d s, d y ) W ( s − , x ) | y − x |≤ r/ . (10.2.12) Proof.
Let ξ t = exp( βω ( V t ( B )) − λ ( β ) νr d t ) and observe that Z R W ( t, x ) ϕ ( x )d x = P [ ξ t ϕ ( B t )] . Then, recalling that ω ( V t ( B )) = R χ s,x ω t (d s d x ), we use Itˆo’s formula [31, Section II.5] for fixed B to get that ξ t = 1 − λνr d Z t ξ s d s + λ Z (0 ,t ] × R ξ s − χ s,x ω (d s d x )= 1 + λ Z (0 ,t ] × R ξ s − χ s,x ¯ ω (d s d x ) , (10.2.13)as almost surely, P -a.s. ξ s = ξ s − a.e.As a difference of two increasing processes, ξ is of finite variation over all bounded timeintervals. Also note that one can get an expression to the measure associated to ξ from thelast equation. By the integration by part formula [33, p.52], ξ t ϕ ( B t ) = ξ ϕ ( B ) + Z t ξ s − d ϕ ( B s ) + Z t ϕ ( B s )d ξ s + [ ξ, ϕ ( B )] t , where [ ξ, ϕ ( B )] t = 0 since ϕ ( B ) is continuous. Applying Itˆo ’s formula on d ϕ ( B ) and thentaking P -expectation (which cancels the martingale term in the Itˆo formula), one obtains by(10.2.13) that P -a.s. Z R W ( t, x ) ϕ ( x )d x = ϕ (0) + 12 Z t P [ ξ s − ∆ ϕ ( B s )]d s + λ Z (0 ,t ] × R P [ ϕ ( B s ) ξ s − χ s,y ]¯ ω (d s d y )= ϕ (0) + 12 Z t Z R ∆ ϕ ( x ) W ( s − , x )d x d s + λ Z (0 ,t ] × R (cid:18)Z R ϕ ( x ) | y − x |≤ r/ W ( s − , x )d x (cid:19) ¯ ω (d s d y ) . To conclude the proof, observe that we can apply Fubini’s theorem to the last integral sincefor all t > P Z (0 ,t ] × R P [ | ϕ ( B s ) | ξ s − χ s,y ] ω (d s d y ) = νe β Z (0 ,t ] × R P [ ξ s − ] P [ | ϕ ( B s ) | χ s,y ]d s d y = νe β r Z t P [ | ϕ ( B s ) | ]d s < ∞ , PPE
Let us first introduce some notations. For any k ≥ s , . . . , s k ∈ R + and x , . . . , x k ∈ R , write s = ( s , . . . , s k ) and x = ( x , . . . , x k ). Let∆ k (0 , t ) = { s ∈ [ u, t ] k | < s < · · · < s k ≤ t } , (10.3.14)be the k -dimensional simplex and ∆ k = ∆ k (0 , We give here the definition of a mild solution to the stochastic heat equation, and we will seehow this leads to an expression of the solution as a Wiener chaos expansion. We first mentionthat it is possible (cf. [34]) to extend the integral over the space time white noise to any squareintegrable function:
Proposition 10.3.1.
There exists an isometry I : L (cid:0) [0 , × R (cid:1) L (Ω , G , P ) verifying:(i) For all measurable set A of [0 , × R , we have I ( A ) = η ( A ) .(ii) For all g ∈ L , the variable I ( g ) is a centered Gaussian variable of variance k g k L ([0 , × R ) . We call I ( g ) the Wiener integral which also writes I ( g ) = R [0 , R R g ( s, x ) η (d s, d x ) . It is said that Z is a mild solution to the stochastic heat equation (10.1.5) if, for all0 ≤ S < T ≤ Z ( T, X ) = Z R ρ ( T − S, X − Y ) Z ( S, Y )d Y + β Z TS Z R ρ ( T − U, X − Y ) Z ( U, Y ) η ( U, Y )d U d Y, (10.3.15)and if for all T ≥ Z ( T, X ) is measurable with respect to the white noise on [0 , T ] × R . Remark 10.3.2.
As a motivation to look at this form of the equation, one can check that if Z ( T, X ) satisfies (10.3.15) with a smooth deterministic function η ( U, Y ) , then Z ( T, X ) is asolution to the SHE (10.1.5) with smooth noise. Remark 10.3.3.
Under some integrability condition, it can be shown that there is a uniquemild solution - up to indistinguishability - to the SHE with Dirac initial condition [11] . Thissolution is continuous in time and space for ( T, X ) ∈ (0 , × R , and it is continuous in T = 0 in the space of distributions. Furthermore, Z β ( T, X ) can be shown to be positive for all T > . Using the initial condition Z (0 , X ) = δ X , we get by iterating equation (10.3.15) that Z ( T, X ) = ρ ( T, X ) + β Z T Z ρ ( T − U, X − Y ) ρ ( U, Y ) η ( U, Y )d U d Y + β Z Z PPE Property 10.3.4. For all k > , there exists a map I k : L (∆ k × R k ) L (Ω , G , P ) , whichhas the following properties:(i) For all g ∈ L (∆ k × R k ) and h ∈ L (∆ j × R j ) , the variable I k ( g ) is centered and P [ I k ( g ) I j ( h )] = δ k,j < g, h > L (∆ k × R k ) . (10.3.16) (ii) The map I k is linear, in the sense that for all square-integrable f, g and reals λ, µ , P -a.s. I k ( λf + µg ) = λ I k ( f ) + µ I k ( g ) . The operator I k is called the multiple Wiener integral , and for g ∈ L (∆ k × R k ) , wealso write I k ( g ) = Z ∆ k Z R k g ( t , x ) η ⊗ k (d t , d x ) . Remark 10.3.5. As a justification of the ”iterated integral” property, it can be shown that themap I k extends to L ([0 , k × R k ) , where it verifies that for all orthogonal family ( g , . . . , g k ) of functions in L ([0 , × R ) : I k k O j =1 g j = k Y j =1 I ( g j ) , (10.3.17) where N denotes the tensor product: ( N kj =1 g j )( s , x ) = Q kj =1 g j ( s j , x j ) . By repeating the above iteration procedure, one gets that: Z ( T, X ) = ρ ( T, X ) + ∞ X k =1 β k Z ∆ k Z R k ρ k ( S , Y ; T, X ) η ⊗ k (d S , d Y ) , (10.3.18)where we have used the notation, for s ∈ ∆ k ( s, t ) and y ∈ R d , ρ k ( s , y ; t, x ) = ρ ( s , y ) k − Y j =1 ρ ( s j +1 − s j , y j +1 − y j ) ρ ( t − s k , x − y k ) . The infinite sum (10.3.18) is called a Wiener chaos expansion . By the covariance structureof the Wiener integrals (10.3.16), all the integrals in the sum are orthogonal and to prove that(10.3.18) converges in L , it suffices to check that (see [1]): ∞ X k =0 k ρ k ( · , · ; T, X ) k L (∆ k × R k ) < ∞ . The ratio ρ k ( s , y ; t,x ) ρ ( t,x ) is the k -steps transition function of a Brownian bridge, starting from(0 , 0) and ending at ( t, x ). From this observation, it is possible to introduce an alternativeexpression of the mild solution Z ( T, X ) of SHE equation, via a Feynman-Kac formula: Z ( T, X ) = ρ ( T, X ) P T,X , (cid:20) : exp : (cid:18) β Z T η ( u, B u )d u (cid:19)(cid:21) , (10.3.19)The Wick exponential : exp : of a Gaussian random variable ξ is defined by: exp( ξ ) := ∞ X k =0 k ! : ξ k : PPE ξ k : notation stands for the Wick power of a random variable (cf.[34]). The integral R T η ( u, B u )d u , on the other hand, is not well defined, and to understand how to go from(10.3.19) to (10.3.18), one should use the following identification: P T,X , " : (cid:18) β Z T η ( u, B u )d u (cid:19) k : = β k k ! Z ∆ k Z R k ρ k ( S , Y ; T, X ) ρ ( T, X ) η ⊗ k (d S d Y ) . From now on, we suppose that Z β ( T, X ) is defined through equation (10.3.18). Integratingover X this equation leads to the definition of the partition function of the continuum polymer: Z β = ∞ X k =0 β k I k ( ρ k ) , (10.3.20)where ρ k is the k -th dimensional Brownian transition function, defined for ( s , x ) ∈ ∆ k × R k by: ρ k ( s , x ) = ρ ( s , x ) k − Y j =1 ρ ( s j +1 − s j , x j +1 − x j ) = P ( B s ∈ d x , . . . , B s k ∈ d x k ) , (10.3.21)with the convention that ρ = 1. The motivation for writing the partition function as in(10.3.20) is that (10.3.18) writes Z β ( T, X ) = P ∞ k =0 β k I k ( ρ k ( · ; T, X )). We want to express W t in a similar way as (10.3.20), this time with Poisson iterated integrals.We give here the basic definitions of these integrals and one can refer to [41] for more details . Definition 10.3.6. For any positive integer k , define the k -th factorial measure ω ( k ) t to bethe point process on ∆ k (0 , t ) × R k , such that, for any measurable set A ⊂ ∆ k (0 , t ) × R k , ω ( k ) t ( A ) = X ( s ,x ) ,..., ( s k ,x k ) ∈ ω t s < ··· For k ≥ and g ∈ L (∆ k (0 , t ) × R k ) , denote the multiple Wiener-Itˆointegral of g as ¯ ω ( k ) t ( g ) := X J ⊂ [ k ] ( − k −| J | Z ∆ k × R k g ( s , x ) ω ( | J | ) t (d s J , d x J ) ν k −| J | d s J c d x J c . (10.3.24) When k = 0 , define ¯ ω (0) t to be the identity on R . Note that for simplicity, we choosed to define here the integrals for functions of the simplex, so that somenormalizing k ! terms and symmetrisation of some objects should be added to match the definitions in [41]. PPE Proposition 10.3.8. For k ≥ , the map ¯ ω t ( k ) can be extended to a map ¯ ω ( k ) t : L (∆ k (0 , t ) × R k ) → L (Ω , G , P ) g ¯ ω ( k ) t ( g ) , which coincides with the above definition of ¯ ω ( k ) t on the functions of L ∩ L (∆ k (0 , t ) × R k ) . Property 10.3.9. (i) For any k ≥ and g ∈ L (∆ k (0 , t ) × R k ) , we have P (cid:2) ¯ ω ( k ) t ( g ) (cid:3) = 0 .(ii) For any k ≥ and l ≥ , g ∈ L (∆ k (0 , t ) × R k ) and h ∈ L (∆ l (0 , t ) × R l ) , the followingcovariance structure holds: P h ¯ ω ( k ) t ( g ) ¯ ω ( l ) t ( h ) i = δ k,l ν k < g, h > L (∆ k (0 ,t ) × R k ) . (10.3.25) (iii) The map ¯ ω ( k ) t is linear, in the sense that for all square-integrable f, g and reals λ, µ , P -a.s. ¯ ω ( k ) t ( λf + µg ) = λ ¯ ω ( k ) t ( f ) + µ ¯ ω ( k ) t ( g ) . Proposition 10.3.10. [22] The renormalized partition function admits the following Wiener-Itˆo chaos expansion: W t = ∞ X k =0 ¯ ω ( k ) t (Ψ k ) , (10.3.26) where the sum converges in L and where, for all s ∈ ∆ k , x ∈ R k and k ≥ , we have set: Ψ k ( s , x ) = λ ( β ) k P " k Y i =1 χ s i ,x i ( B ) , (10.3.27) with the convention that an empty product equals .Sketch of proof. We follow the proof of Lemma 18.9 in [41]. By definition we have that W t = P (cid:2) e βω ( V t ( B ) − tλrν (cid:3) . Hence, assuming that Fubini’s theorem applies to the RHS of (10.3.26), itis enough to show that P × P -almost surely: e βω ( V t ( B )) − tλrν = ∞ X k =0 ¯ ω ( k ) t (cid:16) ( λχ ) ⊗ k (cid:17) , (10.3.28)where, for all s ∈ ∆ k (0 , t ), x ∈ R k , we have defined ( λχ ) ⊗ k ( s , x ) = Q kj =1 λ ( β ) χ s j ,x j ( B ). Then,observe that: ∞ X k =0 ¯ ω ( k ) t (cid:16) ( λχ ) ⊗ k (cid:17) = ∞ X k =0 X J ⊂ [ k ] ( − k −| J | Z ∆ k (0 ,t ) × R k k Y i =1 λχ s i ,x i ω ( | J | ) t (d s J , d x J ) ν k −| J | d s J c d x J c = ∞ X k =0 k X j =0 (cid:18) kj (cid:19) ( − k − j k ! Z [0 ,t ] k × R k k Y i =1 λχ s i ,x i ω ( j ) t (d s [ j ] , d x [ j ] ) ν k − j d s [ j ] c d x [ j ] c = ∞ X j =0 j ! j ! ω ( j ) (cid:0) ( λχ ) ⊗ j (cid:1) ∞ X k = j k − j )! ( − tλνr ) k − j = e − tλrν ∞ X j =0 ω ( j ) (cid:0) ( λχ ) ⊗ j (cid:1) . PPE s , x ) , . . . , ( s N , x N ) with s < · · · < s N be the points of ω that lie in the tube V t ( B ), we get by definition of the ω ( j ) t ’s that ∞ X j =0 ω ( j ) t (cid:0) ( λχ ) ⊗ j (cid:1) = X J ⊂ [ N ] Y i ∈ J (cid:16) e βχ si,xi − (cid:17) = N Y i =1 e βχ si,xi = e βω ( V t ( B )) , where the last equality comes from a telescopic sum (and the convention that an empty productis 1). This implies (10.3.28).To prove convergence in L of the sum in the RHS of (10.3.26), notice that the terms arepairwise orthogonal and verify: P h ¯ ω ( k ) t ( T k W t ) i = λ k Z ∆ k (0 ,t ) × R k P " k Y i =1 χ s i ,x i ( B ) d s d x ≤ λ k k ! Z [0 ,t ] k × R k P " k Y i =1 χ s i ,x i ( B ) d s d x = ( λ tνr ) k k ! , whose sum converges. We now consider parameters β t ∈ R , ν t > r t > t , and we fix aparameter β ∗ ∈ R ∗ . We assume that they verify the following asymptotic relations, as t → ∞ :(a) ν t r t λ ( β t ) ∼ ( β ∗ ) t − / , (b) ν t r t λ ( β t ) → , (c) r t / √ t → . (10.4.29)Suppose for example that r t = ν t = 1. Then, the scaling conditions are equivalent to β t = β ∗ t − / , so that we can see the scaling as a limit from strong disorder ( β > 0) to weakdisorder ( β = 0). For general parameters, one can observe that in dimension d ≥ 3, Theorem3.3.1 implies that there exists a positive constant c ( d ), such that the polymer lies in the L region as soon as νr d +2 λ ( β ) < c ( d ). Since conditions (a) and (c) imply that ν t r t λ ( β t ) → d = 1 should again be interpreted as a crossover between strong and weakdisorder. Remark 10.4.1. The asymptotics are in contrast to the regime of complete localization (9.2.1) ,where, for fixed r , one let νβ → ∞ . The following theorem states that under the above scaling, the P2L and P2P partitionfunctions of the Poisson polymer converge to the one of the continuum polymer: Theorem 10.4.2. Suppose conditions (a), (b) and (c) hold. Then, as t → ∞ : W t ( ω ν t , β t , r t ) law −→ Z β ∗ , (10.4.30) where ω ν t is the Poisson point process with intensity measure ν t d s d x . Moreover, for all S, Y, T, X ∈ [0 , , we have √ tW (cid:16) tS, √ tY ; tT, √ tX ; ω ν t , β t , r t (cid:17) law −→ Z β ∗ ( S, Y ; T, X ) , (10.4.31) PPE where the renormalized P2P partition function from ( S, Y ) to ( T, X ) is defined by W ( s, y ; t, x ; ω, β, r ) = W ( t − s, x − y ; ω, β, r ) ◦ θ s,y , (10.4.32) and similarly for Z β ∗ ( S, Y ; T, X ) . Remark 10.4.3. The √ t term appears here as a renormalization in the scaling of the heatkernel: √ tρ (cid:0) tT, √ tX (cid:1) = ρ ( T, X ) .Sketch of proof. We focus on showing (10.4.30), as the result for the P2P partition functionfollows from the same technique and remark 10.4.3. Let γ t be proportional to the vanishingparameter appearing in scaling relation (b): γ t := ( β ∗ ) − ν t r t λ ( β t ) → . (10.4.33)and we now specify the radius for the indicator χ δs,x ( B ) = | B s − y |≤ δ/ . Introduce the followingtime-depending functions of ∆ k (0 , t ) × R k : φ kt ( s , x ) = γ − kt λ ( β t ) k P " k Y i =1 χ r t / √ ts i ,x i ( B ) . (10.4.34)Note that for all ( s, x ), the diffusive scaling property of the Brownian motion implies that χ r t / √ ts/t,x/ √ t = | B s/t − x/ √ t |≤ r t / √ t law = χ r t s,x . Therefore, using notation e φ kt = φ kt ( · /t, · / √ t ), we see that after simple rescaling, equation(10.3.27) becomes γ kt e φ kt ( s , x ) = λ ( β ) k P " k Y i =1 χ s i ,x i ( B ) . (10.4.35)Hence, Proposition (10.3.10) and equation (10.4.35) lead to the following expression of W t : W t = ∞ X k =0 γ kt ¯ ω ( k ) t (cid:16) e φ kt (cid:17) . (10.4.36)Now, we also define the rescaled functions ˜ ρ kt = ρ k ( · /t, · / √ t ) and we make two claims: • Claim 1: For all k ≥ t → ∞ , φ kt L −→ ( β ∗ ) k ρ k , • Claim 2: As t → ∞ , P ∞ k =0 ( β ∗ ) k γ kt ¯ ω ( k ) t (˜ ρ kt ) law −→ P ∞ k =0 ( β ∗ ) k I k ( ρ k ) = Z β ∗ .Claim 1 follows from the scaling relations and the fact that ε − k P [ Q ki =1 χ εs i ,x i ( B )] → ρ k ( s , x ) as ε → 0. For claim 2, we only present the argument for the convergence in law of the k = 1 termof the sum. The complete argument relies on this case to extend the convergence to all k ≥ ω (1) = ˜ ω and ρ = ρ , we can apply the complex exponential formula (seeequation (2.2.7)) to compute the characteristic function of γ t ¯ ω t (˜ ρ t ). For u ∈ R , we obtain: P h e iuγ t ¯ ω t (˜ ρ t ) i = exp Z [0 ,t ] Z R (cid:0) e iuγ t ρ ( s/t,x/ √ t ) − − iuγ t ρ ( s/t, x/ √ t ) (cid:1) ν t d s d x ! = exp Z [0 , Z R ν t t / (cid:0) e iuγ t ρ ( s,x ) − − iuγ t ρ ( s, x ) (cid:1) d s d x ! . Note that from now on, we will always assume that ω law = ω ν t , although we will drop the superscript notation. PPE ∀ ( s, x ) ∈ [0 , × R , ν t t / (cid:12)(cid:12)(cid:12) e iuγ t ρ ( s,x ) − − iuγ t ρ ( s, x ) (cid:12)(cid:12)(cid:12) ≤ ν t t / γ t u ρ ( s, x ) , This gives L domination since since ρ ∈ L ([0 , × R ) and since relations (a) and (b) implythat ν t γ t ∼ t − / . Moreover, as γ t → 0, the integrand of the above integral converges pointwiseto the function ( s, x ) 7→ − u ρ ( s, x ). Therefore, by dominated convergence, we obtain that as t → ∞ , P h e iuγ t ˜ ω t (˜ ρ t ) i → exp (cid:18) − u k ρ k (cid:19) . This is the Fourier transform of a centered Gaussian random variable of variance k ρ k , whichhas the same law as I ( ρ ), so that indeed ¯ ω (1) t (˜ ρ t ) law −→ I ( ρ ).Now, if X n and Y n are random variables such that Y n law −→ Y and k Y n − X n k −→ 0, then X n law −→ Y . Therefore, to prove that W t law −→ Z β ∗ , it is enough by claim 2 to show that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k =0 γ kt ¯ ω ( k ) t ( ˜ φ kt ) − ∞ X k =0 ( β ∗ ) k γ kt ¯ ω ( k ) t (˜ ρ kt ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) −→ t →∞ . (10.4.37)By Pythagoras’ identity and linearity of ˜ ω ( k ) t , we obtain that the above norm writes: ∞ X k =0 γ kt k ¯ ω ( k ) t (cid:0) φ kt ( · /t, · / √ t ) − ( β ∗ ) k ρ k ( · /t, · / √ t ) (cid:1) k . For all g ∈ L (∆ k × R k ), we get from a substitution of variables that k ¯ ω ( k ) t (cid:0) g ( · /t, · / √ t ) k = ν kt k g ( · /t, · / √ t ) k L (∆ k (0 ,t ) × R k ) = ν kt t k/ k g k L (∆ k × R k ) , so that the above sum is given by ∞ X k =0 γ kt ν kt t k/ k φ kt − ( β ∗ ) k ρ k k L (∆ k × R k ) . Conditions (a) and (b) imply that γ t ν kt t / ∼ 1, so the proof can be concluded by claim 1 andby showing that k φ kt k is dominated by the summable sequence C k k ρ k k , where C = C ( β ∗ ) issome positive constant, so that the dominated convergence theorem applies. Let D ′ ( R ) denote the space of distributions on R , and D (cid:0) [0 , , D ′ ( R ) (cid:1) the space of c`adl`agfunction with values in the space of distributions, equipped with the topology defined in [44].We also define the rescaling of the renormalized P2P partition function (10.2.10): Y t ( T, X ) = ρ ( T, X ) W (cid:16) tT, √ tX ; ω ν t , β t , r t (cid:17) . (10.5.38)The two variables function Y t can be seen as an element of D (cid:0) [0 , , D ′ ( R ) (cid:1) , through themapping Y t : T (cid:0) ϕ R Y t ( T, X ) ϕ ( X )d X (cid:1) . The next theorem states that the rescaledpartition function Y t converges, in terms of processes, to the solution of the stochastic heatequation: PPE Theorem 10.5.1. [22] Suppose that ( β t ) t ≥ is bounded by above. As t → ∞ , the followingconvergence of processes holds: Y t law −→ (cid:0) T 7→ Z β ∗ ( T, · ) (cid:1) , (10.5.39) where the convergence in distribution holds in D (cid:0) [0 , , D ′ ( R ) (cid:1) . For any function F ∈ D ([0 , , D ′ ( R )) and ϕ ∈ D ( R ), set F ( T, ϕ ) := Z F ( T, X ) ϕ ( X )d X. (10.5.40)In order to show tightness of Y t , the tool used in [22] is Mitoma’s criterion [44, 65]: Proposition 10.5.2. Let ( F t ) t ≥ be a family of processes in D ([0 , , D ′ ( R )) . If, for all ϕ ∈D ( R ) , the family T → F t ( T, ϕ ) , t ≥ is tight in the real cadl`ag functions space D ([0 , , R ) ,then ( F t ) t ≥ is tight in D ([0 , , D ′ ( R )) . Then, to prove uniqueness of the limit, one can rely on the following proposition: Proposition 10.5.3 ([44]) . Let ( F t ) t ≥ be a tight family of processes in the space D ([0 , , D ′ ( R )) .If there exists a process F ∈ D ([0 , , D ′ ( R )) such that, for all n ≥ , T , . . . , T n ∈ [0 , and ϕ , . . . , ϕ n ∈ D ( R ) , we have as t → ∞ : ( F t ( T , ϕ ) , . . . , F t ( T n , ϕ n )) law −→ ( F t ( T , ϕ ) , . . . , F t ( T n , ϕ n )) , then F t law −→ F . ibliography [1] Tom Alberts, Konstantin Khanin, and Jeremy Quastel. The continuum directed random polymer. J. Stat. Phys. , 154(1-2):305–326, 2014.[2] Tom Alberts, Konstantin Khanin, and Jeremy Quastel. The intermediate disorder regime fordirected polymers in dimension 1 + 1. Ann. Probab. , 42(3):1212–1256, 2014.[3] Gideon Amir, Ivan Corwin, and Jeremy Quastel. Probability distribution of the free energy of thecontinuum directed random polymer in 1 + 1 dimensions. Comm. Pure Appl. Math. , 64(4):466–537,2011.[4] Antonio Auffinger and Michael Damron. The scaling relation χ = 2 ξ − ALEA Lat. Am. J. Probab. Math. Stat. , 10(2):857–880, 2013.[5] Antonio Auffinger and Michael Damron. A simplified proof of the relation between scaling exponentsin first-passage percolation. Ann. Probab. , 42(3):1197–1211, 2014.[6] M´arton Bal´azs, Jeremy Quastel, and Timo Sepp¨al¨ainen. Fluctuation exponent of theKPZ/stochastic Burgers equation. J. Amer. Math. Soc. , 24(3):683–708, 2011.[7] Erik Bates and Sourav Chatterjee. The endpoint distribution of directed polymers. arXiv , 2016.[8] Quentin Berger and Hubert Lacoin. The high-temperature behavior for the directed polymer indimension 1+ 2. Ann. Inst. Henri Poincar´e Probab. Stat. , 53:430–450, 2017.[9] Pierre Bertin. Free energy for brownian directed polymers in random environment in dimensionone and two. 2008.[10] Pierre Bertin. Very strong disorder for the parabolic Anderson model in low dimensions. Indag.Math. (N.S.) , 26(1):50–63, 2015.[11] Lorenzo Bertini and Nicoletta Cancrini. The stochastic heat equation: Feynman-Kac formula andintermittence. J. Statist. Phys. , 78(5-6):1377–1401, 1995.[12] Lorenzo Bertini and Giambattista Giacomin. Stochastic Burgers and KPZ equations from particlesystems. Comm. Math. Phys. , 183(3):571–607, 1997.[13] Ren´e A. Carmona and S. A. Molchanov. Parabolic Anderson problem and intermittency. Mem.Amer. Math. Soc. , 108(518):viii+125, 1994.[14] Subrahmanyan Chandrasekhar, Mark Kac, and R. Smoluchowski. Marian Smoluchowski: his lifeand scientific work. PWN—Polish Scientific Publishers, Warsaw, 2000.[15] Sourav Chatterjee. The universal relation between scaling exponents in first-passage percolation. Ann. of Math. (2) , 177(2):663–697, 2013.[16] John MC Clark. The representation of functionals of brownian motion by stochastic integrals. TheAnnals of Mathematical Statistics , pages 1282–1295, 1970.[17] Francis Comets. Directed polymers in random environments. ´Ecole d’ ´Et´e de Probabilit´es de Saint-Flour XLVI – 2016. Cham: Springer, 2017.[18] Francis Comets and Nobuo Yoshida. Some new results on brownian directed polymers in randomenvironment. RIMS Kokyuroku , 1386:50–66, 2004.[19] Francis Comets and Nobuo Yoshida. Brownian directed polymers in random environment. Comm.Math. Phys. , 254(2):257–287, 2005. PPE [20] Francis Comets and Nobuo Yoshida. Localization transition for polymers in Poissonian medium. Comm. Math. Phys. , 323(1):417–447, 2013.[21] Ivan Corwin. Kardar-parisi-zhang universality. Notices of the AMS , 63(3):230–239, 2016.[22] Cl´ement Cosco. The intermediate disorder regime for brownian directed polymers in poisson envi-ronment. arXiv:1804.09571 , 2018.[23] Amir Dembo and Ofer Zeitouni. Large deviations techniques and applications , volume 38 of Appli-cations of Mathematics (New York) . Springer-Verlag, New York, second edition, 1998.[24] Frank den Hollander. Random polymers , volume 1974 of Lecture Notes in Mathematics . Springer-Verlag, Berlin, 2009. Lectures from the 37th Probability Summer School held in Saint-Flour, 2007.[25] Monroe Donsker and Srinivasa Varadhan. Asymptotics for the wiener sausage. Communicationson Pure and Applied Mathematics , 28(4):525–565, 1975.[26] Richard Durrett. Probability: theory and examples . Duxbury Press, Belmont, CA, second edition,1996.[27] Giambattista Giacomin. Random polymer models . Imperial College Press, London, 2007.[28] C Douglas Howard. Lower bounds for point-to-point wandering exponents in euclidean first-passagepercolation. Journal of applied probability , 37(4):1061–1073, 2000.[29] C Douglas Howard and Charles M Newman. Euclidean models of first-passage percolation. Prob-ability Theory and Related Fields , 108(2):153–170, 1997.[30] Elton P. Hsu and Karl-Theodor Sturm. Maximal coupling of euclidean brownian motions. Com-munications in Mathematics and Statistics , 1(1):93–104, Mar 2013.[31] N. Ikeda and S. Watanabe. Stochastic differential equations and diffusion processes. Second Edition .Kodansha scientific books. North-Holland, 1989.[32] Kiyosi Itˆo. Multiple wiener integral. Journal of the Mathematical Society of Japan , 3(1):157–169,1951.[33] Jean Jacod and Albert Shiryaev. Limit theorems for stochastic processes, 2nd edition , volume 288.Springer Science & Business Media, 2003.[34] Svante Janson. Gaussian Hilbert Spaces . Cambridge Tracts in Mathematics. Cambridge UniversityPress, 1997.[35] Ioannis Karatzas and Steven E. Shreve. Brownian motion and stochastic calculus. 2nd ed. Springer-Verlag (New York), 1991.[36] Mehran Kardar, Giorgio Parisi, and Yi-Cheng Zhang. Dynamic scaling of growing interfaces. Physical Review Letters , 56(9):889, 1986.[37] Davar Khoshnevisan. Analysis of stochastic partial differential equations. , volume 119 of CBMS Re-gional Conference Series in Mathematics . Published for the Conference Board of the MathematicalSciences, Washington, DC, 2014.[38] Joaquim Krug and Herbert Spohn. Kinetic roughening of growing surfaces (in: Solids far fromequilibrium, Godr`eche ed.) , pages 477–582. Cambridge University Press, Cambridge, 1992.[39] Hubert Lacoin. New bounds for the free energy of directed polymers in dimension 1 + 1 and 1 + 2. Comm. Math. Phys. , 294(2):471–503, 2010.[40] G¨unter Last. Stochastic Analysis for Poisson Processes , pages 1–36. Springer International Pub-lishing, Cham, 2016. PPE [41] G¨unter Last and Mathew Penrose. Lectures on the Poisson process. Cambridge University Press(IMS Textbook), 2017.[42] G¨unter Last and Mathew D. Penrose. Martingale representation for poisson processes with appli-cations to minimal variance hedging. Stochastic Processes and their Applications , 121(7):1588 –1606, 2011.[43] C. Licea, C. M. Newman, and M. S. T. Piza. Superdiffusivity in first-passage percolation. Probab.Theory Related Fields , 106(4):559–591, 1996.[44] Itaru Mitoma. Tightness of probabilities on c ([0, 1]; y’) and d ([0, 1]; y’). The Annals of Probability ,pages 989–999, 1983.[45] S. A. Molchanov, A. A. Ruzmaikin, D. D. Sokoloff, and Ya. B. Zeldovich. Intermittency, diffusionand generation in a nonstationary random medium . Reviews in Mathematics and MathematicalPhysics, 15/1. Cambridge Scientific Publishers, Cambridge, 2014.[46] Gregorio R. Moreno Flores. On the (strict) positivity of solutions of the stochastic heat equation. Ann. Probab. , 42(4):1635–1643, 07 2014.[47] Gregorio R. Moreno Flores, Timo Sepp¨al¨ainen, and Benedek Valk´o. Fluctuation exponents fordirected polymers in the intermediate disorder regime. Electron. J. Probab. , 19:no. 89, 28, 2014.[48] Carl Mueller. On the support of solutions to the heat equation with noise. Stochastics StochasticsRep. , 37(4):225–245, 1991.[49] Daniel Ocone. Malliavin’s calculus and stochastic integral representations of functional of diffusionprocesses. Stochastics: An International Journal of Probability and Stochastic Processes , 12(3-4):161–185, 1984.[50] Firas Rassoul-Agha and Timo Sepp¨al¨ainen. Process-level quenched large deviations for randomwalk in random environment. In Annales de l’institut Henri Poincar´e (B) , volume 47, pages 214–242, 2011.[51] Daniel Revuz and Marc Yor. Continuous martingales and Brownian motion, 3d edition , volume293. Springer Science & Business Media, 2005.[52] R. Tyrrell Rockafellar. Convex analysis . Princeton Mathematical Series, No. 28. Princeton Univer-sity Press, Princeton, N.J., 1970.[53] L. Chris Rogers and David Williams. Diffusions, Markov processes and Martingales, vol 2: Itocalculus , volume Reprint of the second (1994) edition. Cambridge University Press, 2000.[54] Carles Rovira and Samy Tindel. On the brownian-directed polymer in a gaussian random environ-ment. Journal of Functional Analysis , 222:178–201, 2005.[55] Timo Sepp¨al¨ainen. Scaling for a one-dimensional directed polymer with boundary conditions. Ann.Probab. , 40:19–73, 2012.[56] Timo Sepp¨al¨ainen and Benedek Valk´o. Bounds for scaling exponents for a 1+1 dimensional directedpolymer in a Brownian environment. ALEA Lat. Am. J. Probab. Math. Stat. , 7:451–476, 2010.[57] Yuichi Shiozawa. Central limit theorem for branching Brownian motions in random environment. J. Stat. Phys. , 136(1):145–163, 2009.[58] Yuichi Shiozawa. Localization for branching Brownian motions in random environment. TohokuMath. J. (2) , 61(4):483–497, 2009.[59] Yakov G. Sinai. A remark concerning random walks with random potentials. Fund. Math. ,147(2):173–180, 1995. PPE [60] Dietrich Stoyan, Wilfrid Kendall, and Joseph Mecke. Stochastic geometry and its applications .Akademie-Verlag, Berlin, 1987.[61] Alain-Sol Sznitman. Brownian motion, obstacles and random media . Springer Monographs inMathematics. Springer-Verlag, Berlin, 1998.[62] Craig A Tracy and Harold Widom. Level-spacing distributions and the airy kernel. Communicationsin Mathematical Physics , 159(1):151–174, 1994.[63] Vincent Vargas. A local limit theorem for directed polymers in random media: the continuous andthe discrete case. Ann. Inst. H. Poincar´e Probab. Statist. , 42(5):521–534, 2006.[64] Vincent Vargas. Strong localization and macroscopic atoms for directed polymers. Probabilitytheory and related fields , 138(3-4):391–410, 2007.[65] John B. Walsh. An introduction to stochastic partial differential equations. In ´Ecole d’´et´e deprobabilit´es de Saint-Flour, XIV—1984 , volume 1180 of Lecture Notes in Math. , pages 265–439.Springer, Berlin, 1986.[66] Mario V. W¨uthrich. Scaling identity for crossing Brownian motion in a Poissonian potential. Probab.Theory Related Fields , 112(3):299–319, 1998.[67] Mario V. W¨uthrich. Superdiffusive behavior of two-dimensional Brownian motion in a Poissonianpotential. Ann. Probab. , 26(3):1000–1015, 1998.[68] Mario V. W¨uthrich. Geodesics and crossing Brownian motion in a soft Poissonian potential. Ann.Inst. H. Poincar´e Probab. Statist. , 35(4):509–529, 1999.[69] Mario V. W¨uthrich. Numerical bounds for critical exponents of crossing Brownian motion.