Uniqueness and ergodicity of stationary directed polymers on Z 2
aa r X i v : . [ m a t h . P R ] A p r UNIQUENESS AND ERGODICITY OF STATIONARYDIRECTED POLYMERS ON Z CHRISTOPHER JANJIGIAN AND FIRAS RASSOUL-AGHA
Abstract.
We study the ergodic theory of stationary directed nearest-neighbor polymer modelson Z , with i.i.d. weights. Such models are equivalent to specifying a stationary distribution onthe space of weights and correctors that satisfy certain consistency conditions. We show thatfor prescribed weight distribution and corrector mean, there is at most one stationary polymerdistribution which is ergodic under the e or e shift. Further, if the weights have more than twomoments and the corrector mean vector is an extreme point of the superdifferential of the limitingfree energy, then the corrector distribution is ergodic under each of the e and e shifts. Introduction
Directed polymers with bulk disorder were introduced in the statistical physics literature by Huseand Henley [17] in 1985 to model the domain wall separating the positive and negative spins in theferromagnetic Ising model with random impurities. These models have been the subject of intensestudy over the past three decades; see the recent surveys [7, 8, 12, 14]. Much of the reason forthe interest in these models is due to the conjecture that under mild hypotheses, such models aremembers of the Kardar-Parisi-Zhang (KPZ) universality class, which is an extremely wide-rangingclass of models believed to have the same statistical and structural properties. See [9, 10, 16, 24, 25]for recent surveys.Much like characterization of the Guassian universality in terms of variants of the central limittheorem (CLT), the KPZ class is characterized by its scaling exponents and limit distributions.In the context of the directed polymer models we study in this paper, the conjecture is that theappropriately centered and normalized finite volume free energy converges to a limit distributionwhich is independent of the random environment that the polymer lives in. The KPZ scalingtheory [20, 22] predicts that the fluctuations around the limiting free energy are of the order of thecube root of the size of the volume, in contrast to the square root in the classical CLT. Moreover,the limiting distribution is not Gaussian. The effect of the environment is felt only through thecentering and normalizing constants, which play a role similar to that of the mean and standarddeviation in the CLT.The scaling theory also predicts the values of these two non-universal constants if one has acomplete description of the spatially-ergodic and temporally-stationary measures for the polymermodel. See [28] for an example of this computation in the setting of the semi-discrete polymermodel introduced by O’Connell and Yor in [23]. In the mathematics literature, stationary polymermodels have been studied primarily in the context of solvable models, which have product-formstationary distributions. The first such solvable stationary polymer model was the aforementionedmodel due to O’Connell and Yor. The first discrete model was introduced by Sepp¨al¨ainen in [27].
Date : April 11, 2019.2000
Mathematics Subject Classification.
Key words and phrases. cocycle, corrector, directed polymer, Gibbs measure, stationar polymer.C. Janjigian was partially supported by a postdoctoral grant from the Fondation Sciences Math´ematiques de Pariswhile working at Universit´e Paris Diderot.F. Rassoul-Agha was partially supported by National Science Foundation grants DMS-1407574 and DMS-1811090.
See also the models introduced in [3, 11, 29, 30] and studied further in [2, 5, 6]. Such models remainto date the only polymer models for which the KPZ universality conjectures have been verified.In the present paper, we investigate the ergodic theory of stationary directed polymers on thelattice Z with general i.i.d. weights and nearest-neighbor steps. The solvable model in [27] is anexample of the type of model we study. Specializing our results to the case where the limiting freeenergy is everywhere differentiable, we show that the ergodic distributions form a one-parameterfamily, indexed by the derivative of the free energy. This differentiability is satisfied in the modelin [27] and is conjectured to hold generally. As a consequence, our results imply that the ergodicmeasures constructed in [27] are the only such measures in that model, which verifies that thehypotheses of the scaling theory are satisfied. More generally, we give conditions under which onecan conclude that a stationary distribution is ergodic as well as conditions under which an ergodicmeasure is unique.Apart from being fundamental objects for the study of the scaling theory, classifying stationaryergodic polymer measures is important for addressing several other questions. We mention two.Our results on the classification of stationary and ergodic polymer measures can be reformulatedin terms of a characterization of the stationary and ergodic global physical solutions to a discreteversion of the viscous stochastic Burgers equation B t U “ B xx U ` B x U ` B x W , where W is space-time white noise. This connection is the focus of our companion paper [19].In the language of statistical mechanics, stationary directed polymer measures are in correspon-dence with shift-covariant semi-infinite Gibbs measures which are consistent with the quenchedpoint-to-point polymer measures. For a discussion of this point of view, we refer the reader to [18].We close this introduction by giving an outline of the rest of the paper. We introduce polymermeasures and stationary polymer measures in Section 2 then state our main results in Section 3.In Section 4 we prove some preliminary lemmas and motivate the setting of Section 5, where weprove some auxiliary results that are then used in the proofs of the main theorems in Section 6.2. Setting
Random polymer measures.
Let Ω “ R Z and equip it with the product topology andproduct Borel σ -algebra F . A generic point in Ω will be denoted by ω . Let t ω x p ω q : x P Z u bethe natural coordinate projections . The number ω x models the energy stored at site x and is calledthe weight or potential or environment at x . Define the natural shift maps T z : Ω Ñ Ω , z P Z ,by ω x p T z ω q “ ω x ` z p ω q . We are given a probability measure P on p Ω , F q such that t ω x : x P Z u are i.i.d. under P and E r| ω |s ă 8 .Let Π u,v be the set of up-right paths, i.e. paths in Z with steps in t e , e u , from u to v . For m ď n in Z Y t˘8u we write x m,n to denote a path p x m , x m ´ , . . . , x n q and we will use the conventionthat x k ¨ p e ` e q “ k .Given the weights, the quenched point-to-point polymer measures are probability measures onup-right paths between two fixed sites in which the probability of a path is proportional to theexponential of its energy: Q ωu,v p x m,n q “ e ř nk “ m ` ω xk Z ωu,v , x m,n P Π u,v , v ě u. (2.1)Here, Z ωu,v is the quenched point-to-point partition function given by Z ωu,v “ ÿ x m,n P Π u,v e ř nk “ m ` ω xk , v ě u, (2.2)with the convention that an empty sum equals 0. (Thus, Z ωu,u “ Z ωu,v “ v ě u .) TATIONARY DIRECTED POLYMERS 3
A computation shows that the point-to-point measure is a backward Markov chain starting at v , taking steps t´ e , ´ e u , and with absorption at u . If we define B u p x, y, ω q “ log Z ωu,y ´ log Z ωu,x , x, y ě u, then the transition probabilities of this Markov chain are given by ~π u p x, x ´ e i , ω q “ e ω x Z ωu,x ´ e i Z ωu,x “ e ω x ´ B u p x ´ e i ,x,ω q , u ď x ď v, x ‰ u, i P t , u . (2.3)Note that if x “ u ` ke i , k P N , then the chain takes ´ e i steps until it reaches u . The processes B u satisfy the (rooted) cocycle property B u p x, y, ω q ` B u p y, z, ω q “ B u p x, z, ω q , x, y, z ě u. (2.4)Since Z ωu,v satisfies the recurrence Z u,x “ e ω x p Z u,x ´ e ` Z u,x ´ e q , x ´ u P N , (2.5)we see that B u also satisfies the recovery property e ´ B u p x ´ e ,x,ω q ` e ´ B u p x ´ e ,x,ω q “ e ´ ω x , x ´ u P N . (2.6)Note also that B u p x, y, T z ω q “ B u ` z p x ` z, y ` z, ω q (2.7)and that if x, y ď v , then B u p x, y, ω q is a function of t ω z : z ď v u and is hence independent of t ω z : z ď v u .A stationary polymer measure is one that retains the properties (2.4), (2.6), (2.7), and the aboveindependence structure, but without a reference to the roots u and u ` z . This leads us to thenotion of corrector distributions.2.2. Corrector distributions.
Extend the measurable space p Ω , F q to p Ω , F q where Ω “ R Z ˆ R Z ˆ Z , equipped with the product topology, and F is the product Borel σ -algebra. Now, ω willdenote a generic point in Ω and t ω x p ω q : x P Z u and t B p x, y, ω q : x, y P Z u denote the naturalcoordinate projections. The natural shift maps are now given by T z : Ω Ñ Ω, z P Z , with ω x p T z ω q “ ω x ` z p ω q and B p x, y, T z ω q “ B p x ` z, y ` z, ω q . We will abuse notation and keep using ω x and T z to denote, respectively, the natural projections and shifts on the original space Ω .We say that a probability measure P on p Ω , F q is a stationary future-independent L correctordistribution with Ω -marginal P if it satisfies the following:I. Distributional properties: for all x, y, z P Z (a) Prescribed marginal: the Ω -marginal is P ,(b) Stationarity: P is invariant under T z ,(c) Integrability: E r| B p x, y q|s ă 8 ,(d) Future-independence: for any down-right path y “ p y k q k P Z , i.e. y k ` ´ y k P t e , ´ e u , t B p x, y, ω q : D v P y : x, y ď v u and t ω z : z ď v, @ v P y u are independent.II. Almost sure properties: for P -almost every ω and all x, y, z P Z (e) Cocycle: B p x, y q ` B p y, z q “ B p x, z q ,(f) Recovery: e ´ B p x ´ e ,x q ` e ´ B p x ´ e ,x q “ e ´ ω x .We say P is ergodic under T z (or T z -ergodic) if P p A q P t , u for all sets A P F such that T z A “ A .As it is customary with probability notation (and was already done above), we will often omit the ω from the arguments of B p x, y q and ω x . A function B satisfying property (e) is called a cocyle .If it also satisfies (f) then it is called a corrector . Our use of the word “corrector” comes from ananalogy with stochastic homogenization. See e.g. [1, page 467]. The recovery equation (f) is theanalogue of (3.4) in that paper. C. JANJIGIAN AND F. RASSOUL-AGHA
Stationary polymer measures from corrector distributions. A stationary polymermeasure is given by first specifying a stationary future-independent corrector distribution P withan i.i.d. Ω -marginal P . Given a realization of the environment ω , the quenched polymer measure Q ωv rooted at v P Z is a Markov chain starting at v and having transition probabilities ~π p x, x ´ e i , ω q “ e ω x ´ B p x ´ e i ,x,ω q , x P Z , i P t , u . (2.8)Observe that ~π p x, x ´ e i , ω q “ ~π p , ´ e i , T x ω q . Hence, stationary polymer measures are in factexamples of the familiar model of a random walk in a stationary random environment (RWRE).The quenched point-to-point measure (2.1) can also be viewed as a forward Markov chain startingat u , taking steps t e , e u , with absorption at v and transitions ~π v p x, x ` e i , ω q “ e ω x ` ei Z ωx ` e i ,v Z ωx,v “ e ω x ´ B v p x,x ` e i q , u ď x ď v, x ‰ v, i P t , u , (2.9)where now B v p x, y, ω q “ log Z ωx,v ´ log Z ωy,v ` ω x ´ ω y . This structure also leads to stationary polymermeasures that are stationary (forward) RWREs with steps t e , e u and whose transitions are of theform ~π p x, x ` e i , ω q “ e ω x ´ B p x,x ` e i ,ω q , i P t , u , where B is an L stationary corrector but with therecovery equation replaced by e ´ ω x “ e ´ B p x,x ` e q ` e ´ B p x,x ` e q and future-independence replacedby past-independence (defined in the obvious way). The two points of view are in fact equivalentdue to the symmetry of P with respect to reflections of the axes.It should be noted that although the weights t ω x : x P Z u are i.i.d. under P , the transitions t ~π p x, x ´ e q : x P Z u (and t ~π p x, x ` e q : x P Z u ) are highly correlated, causing the paths of theRWREs to be superdiffusive with a 2 { Stationary polymer measures with boundary.
Another, perhaps more familiar, way ofintroducing stationary polymer measures comes by considering solutions to the recursion (2.5), butwith appropriate boundary conditions. This is how these measures were introduced in the studyof solvable models mentioned in the introduction. We explain in this section how this viewpoint isthe same as the one via the framework of corrector distributions.Given a stationary future-independent corrector distribution P with an i.i.d. Ω -marginal P , adown-right path y “ y ´8 , with y m ¨ p e ´ e q “ m for m P Z , and a point u P y , define the quenched path-to-point partition functions Z y ,ωu,v “ ÿ x m,n P Π y ,v e B p u,x m q` ř nk “ m ` ω xk . (2.10)Here, Π y ,v is the set of up-right paths x m,n that start at a point x m P y , exit y right away, i.e. x m ` R y , and end at x n “ v . Recall that an empty sum is 0. If v P y , then Π y ,v consists of asingle path and Z y ,ωu,v “ e B p u,v q .The cocycle and recovery properties (e) and (f) imply that e B p u,x q satisfies the same recurrencerelation (2.5) as Z y ,ωu,x . Since the two also match for x P y we deduce that Z y ,ωu,x “ e B p u,x q for all x for which Π y ,x “ ∅ . In particular, this definition is independent of the boundary y and gives astationary field t e B p u,v q : u, v P Z u of point-to-point partition functions. Also, this explains why B is called a corrector: it corrects the potential t ω x : x P Z u , turning the superadditive log Z ωu,v intoan additive cocycle log Z y ,ωu,v . This is a key idea in stochastic homogenization theory. For more, seefor example Section 2 of [21].The corresponding quenched path-to-point polymer measure is given by Q y ,ωu,v p x m,n q “ e B p u,x m q` ř nk “ m ` ω xk Z y ,ωu,v , x m,n P Π y ,v . TATIONARY DIRECTED POLYMERS 5 Q y ,ωu,v is the distribution of a Markov chain starting at v and having transition probabilities ~π p x, x ´ e i , ω q “ e ω x Z y ,ωu,x ´ e i Z y ,ωu,x “ e ω x ´ B p x ´ e i ,x,ω q , x P Z , i P t , u until reaching y . In other words, this is exactly the quenched distribution Q ωv , until absorption at y and the path-to-point polymer measure is exactly the stationary polymer measure introducedabove.One can also go in the other direction: starting from a stationary model with boundary we candefine a corrector distribution. More precisely, suppose we are given a boundary down-right path y “ y ´8 , with y m ¨ p e ´ e q “ m for m P Z and a point u P y . Abbreviate I ` y “ t z P Z : z ď v, @ v P y u . Equip Ω y “ R I ` y ˆ R y with the product topology and Borel σ -algebra and denote thenatural projections of an element ω P Ω y by ω z , z P I ` y , and ¯ ω v , v P y . Suppose we are given aprobability measure P on Ω y under which t ω z : z P I ` y u and t ¯ ω v : v P y u are independent and suchthat the distribution of t ω z : z P I ` y u is the same under P as under P .Let m “ u ¨ p e ´ e q , so that u “ y m . For m P Z let B p u, x m q “ ř m ´ i “ m ¯ ω i for m ě m and B p u, x m q “ ´ ř m ´ i “ m ¯ ω i for m ď m . Define the path-to-point partition function Z y ,ωu,v , v P I ` y Y y ,by (2.10). The probability measure P is said to be a stationary polymer model with boundary y if the distribution of t ω v ` z , Z y ,ωu,y ` z { Z y ,ωu,u ` z : v P I ` y , y P I ` y Y y u , induced by P , does not depend on z P Z ` .For x, y P I ` y Y y let B y p x, y q “ log Z y ,ωu,y ´ log Z y ,ωu,x . Then the above definition is equivalent to saying that the distribution of t ω v ` z , B y p x ` z, y ` z q : v P I ` y , x, y P I ` y Y y u , induced by P , is the same for all z P Z ` . Kolmogorov’s consistency theorem allows then to extend P to a probability measure P on p Ω , F q and a few direct computations check that P is a stationaryfuture-independent corrector distribution with Ω -marginal P .2.5. The stationary log-gamma polymer.
As mentioned earlier, stationary polymer measureswith boundaries were a crucial tool in the study of solvable models. In this section we recall theexample of Sepp¨al¨ainen’s log-gamma polymer [27], which fits our setting. Related models includethe stationary semi-discrete model in [23], where the boundary p´8 ,
8q ˆ t u was used, whichis analogous to y ´8 , “ Z e , and the models studied in [3, 11, 30], where y ´8 , “ Z ` e and y , “ Z ` e was used. In all of these models, the reference point is taken to be u “ θ ą W θ denote the distribution of a random variable X such that e ´ X is gamma-distributed with scale parameter 1 and shape parameter θ . Let W θ denote the distribution of ´ X .The log-gamma polymer is the directed polymer measure on Z with P being the product measure W b Z ρ for some ρ ą y ´8 , “ Z ` e and y , “ Z ` e and the origin point u “
0. Then I ` y “ N . Fix θ P p , ρ q and let P be the product probability measure W b N ρ b W b N e θ b W b N e ρ ´ θ .The path-to-point partition functions Z y ,ω ,x , x P Z ` , can be computed inductively by the equations Z y ,ω ,x “ e ω x p Z y ,ω ,x ´ e ` Z y ,ω ,x ´ e q , x P N , and the initial conditions Z y ,ω , “ , Z y ,ω ,me “ e ř m ´ i “ ¯ ω ie and Z y ,ω ,me “ e ´ ř mi “ ¯ ω ie , m P Z ` . C. JANJIGIAN AND F. RASSOUL-AGHA
Equivalently, B y p x, x ` e i q , i P t , u , x P Z ` , are computed inductively using e B y p x ´ e ,x q “ e ω x ` ` e B y p x ´ e ´ e ,x ´ e q e ´ B y p x ´ e ´ e ,x ´ e q ˘ ,e B y p x ´ e ,x q “ e ω x ` ` e B y p x ´ e ´ e ,x ´ e q e ´ B y p x ´ e ´ e ,x ´ e q ˘ , (2.11)for x P N , and the initial conditions B y p me , p m ` q e q “ ¯ ω me and B y p me , p m ` q e q “´ ¯ ω p m ` q e , m P Z ` . Compare to (3.2) in [27]. Then B y p x, y q , x, y P Z ` , are computed viathe cocycle property that B y satisfies. The Burke property [27, Theorem 3.3] implies that P isstationary in the sense of the previous section.Alternatively, one can use the boundary path y ´8 , “ Z ` e and y , “ Z ` e and then I ` y “ Z ˆ N and P would be the product measure W b I ` y ρ b W b y θ . The partition functions Z y ,ω ,x , x P Z ˆ Z ` ,are now computed inductively by the equations Z y ,ω ,x “ ÿ m “ e ř mi “ ω x ´ ie Z y ,ω ,x ´ e ´ me , x P Z ˆ N , and the initial conditions Z y ,ω , “ , Z y ,ω ,me “ e ř m ´ i “ ¯ ω ie and Z y ,ω , ´ me “ e ´ ř mi “ ¯ ω ´ ie , m P Z ` . Equivalently, B y p x, x ` e i q , i P t , u , x P Z ˆ Z ` are computed inductively using e B y p x ´ e ,x q “ e ω x ´ ` ÿ m “ m ź i “ e ω x ´ ie ´ B y p x ´ e ´ ie ,x ´ e ´p i ´ q e q ¯ , (2.12) e B y p x ´ e ,x q “ e ω x ` ` e B y p x ´ e ´ e ,x ´ e q e ´ B y p x ´ e ´ e ,x ´ e q ˘ , for x P Z ˆ N , and the initial conditions B y p me , p m ` q e q “ ¯ ω me , m P Z . Analysis of thegeneral polymer model with boundary conditions of this type plays a key role in our analysis. Seein particular the discussion in Section 5.By the uniqueness in Lemma 5.2, the distribution of t B y p x, x ` e i q : x P Z ` u in this constructionis the same as the one in the above construction. Consequently, P is again stationary in the senseof the previous section. See Lemma 5.7 for the details of this argument.3. Main results
Consider a stationary future-independent L corrector distribution P with Ω -marginal P . Byshift-invariance and the cocycle property, E r B p , x ` y qs “ E r B p , x qs ` E r B p x, x ` y qs “ E r B p , x qs ` E r B p , y qs , for all x, y P Z . Hence, there exists a unique vector m P P R , called the mean vector , such that E r B p , x qs “ x ¨ m P for all x P Z . In particular, m P ¨ e i “ E r B p , e i qs .Our first main result is on the uniqueness of ergodic corrector distributions with prescribed i.i.d.Ω -marginal and mean m P ¨ e i . Theorem 3.1.
Fix an i.i.d. probability measure P on p Ω , F q with L weights. Fix a number α P R . Fix i P t , u . There is at most one stationary T e i -ergodic future-independent L correctordistribution P with Ω -marginal P and such that m P ¨ e i “ α . In terms of stationary polymers with boundary y “ Z e , this theorem says that, in general,the T e -ergodic stationary distributions form a one parameter family, indexed by the mean of theboundary weights. The formulation in terms of corrector distributions allows us to extend thisresult to more general boundary geometries.We next turn to the question of determining which values of the parameter m P admit ergodicstationary polymers. To state our second result we need a few more definitions and some more TATIONARY DIRECTED POLYMERS 7 hypotheses. Recall the point-to-point partition functions (2.2). Assume that E r| ω | p s ă 8 for some p ą
2. Then Theorem 2.2(a), Remark 2.3, and Theorems 2.4, 2.6(b), and 3.2(a) of [26] imply thatthere exists a deterministic continuous 1-homogenous concave function Λ P : R ` Ñ R such that P -almost surely n ´ log Z ω , t nξ u ÝÑ n Ñ8 Λ P p ξ q for all ξ P R ` .(3.1)Given ξ P p , , let B Λ P p ξ q “ b P R : b ¨p ξ ´ ζ q ď Λ P p ξ q ´ Λ P p ζ q , @ ζ P R ` ( (3.2)denote the superdifferential of Λ P at ξ . This is a convex set. Let ext B Λ P p ξ q denote its extremepoints. If ξ P p , then Λ P is differentiable at ξ if and only if B Λ P p ξ q “ ext B Λ P p ξ q “ t ∇ Λ P p ξ qu . (3.3)Otherwise, ext B Λ P p ξ q consists of exactly two points (see Lemma 4.6(c) in [18]). It is conjecturedthat Λ P is differentiable on p , and then (3.3) holds for all ξ P p , . Note that due to thehomogeneity of Λ P , B Λ P p ξ q “ B Λ P p cξ q for all ξ P p , and c ą Lemma 3.2. If P is a stationary future-independent corrector distribution with an Ω -marginalgiven by i.i.d. L p weights, p ą , then m P P B Λ P p ξ q for some ξ P t te ` p ´ t q e : 0 ă t ă u “s e , e r . The second is Theorem 4.7 in that paper and gives existence of corrector distributions for eachsuch mean.
Lemma 3.3.
For each b with the property that b P B Λ P p ξ q for some ξ Ps e , e r and for eachprobability measure P on Ω under which the weights are i.i.d. and in L p for some p ą , thereexists a stationary future-independent L corrector distribution P with Ω -marginal P such that m P “ b . Let C P denote the collection of all stationary future-independent L corrector distributions withΩ -marginal P . In words, the last lemma says that as P varies over C P , its mean vector m P spansall of Ť ξ Ps e ,e r B Λ P p ξ q . The next result is an immediate consequence of Lemmas 4.7(b), 4.7(d),and C.1 in [18]. It says that in fact as P spans C P each coordinate of m P spans p E r ω s , . Lemma 3.4.
Fix an i.i.d. probability measure P on p Ω , F q with L p weights, for some p ą .Then t m P : P P C P u is a closed curve in R and for each i P t , u , t m P ¨ e i : P P C P u “ p E r ω s , . Our second main result in this paper gives a convenient tool which allows us to identify ergodicstationary corrector distributions.
Theorem 3.5.
Fix an i.i.d. probability measure P on p Ω , F q with L p weights, for some p ą .Suppose P is a stationary future-independent corrector distribution with Ω -marginal P . If m P P ext B Λ P p ξ q for some ξ Ps e , e r , then P is ergodic under T e and T e . In principle, this result leaves open the possibility that there could be ergodic stationary distri-butions with mean vectors which are not extreme points of the superdifferential of the free energy.Nevertheless, in our setting, it is expected that Λ P is differentiable, in which case every elementof the superdifferential would be extreme. In particular, under this hypothesis, for each ξ Ps e , e r ,Lemma 3.3 furnishes a future-independent L corrector distribution P ξ with Ω -marginal P suchthat m P “ ∇ Λ P p ξ q . Then, under the hypothesis of differentiability, using Theorems 3.1 and 3.5we have the following complete characterization of all ergodic stationary polymer meausures. C. JANJIGIAN AND F. RASSOUL-AGHA
Corollary 3.6.
Fix an i.i.d. probability measure P on p Ω , F q with L p weights, for some p ą .Assume Λ P is differentiable on p , . Then for i P t , u , the collection of T e i -ergodic stationarycorrector distributions is exactly given by t P ξ : ξ Ps e , e ru . In particular, for each ξ Ps e , e r and each i P t , u , P ξ is the unique T e i -ergodic future-independent corrector distribution with Ω -marginal P and such that m P “ ∇ Λ P p ξ q . As a consequence, the above and Lemma 3.4 imply that the one-parameter family of ergodicmeasures constructed in [27] is unique and assumption (2-6) of the scaling theory in [28] is satisfied.4.
Preliminaries
In this section, we motivate the main tool in the proofs of the main theorems 3.1 and 3.5, whichwe will call the update map
Φ.Given B : Z ˆ Z Ñ R define the random variables V n,k “ e ω p n,k q , X n,k “ e B pp n,k ´ q , p n ` ,k ´ qq ,and Y n,k “ e B pp n,k ´ q , p n,k qq´ ω p n,k q for n, k P Z . The next lemma rewrites the corrector property interms of this notation. Compare (4.1) with (2.11). Lemma 4.1. B is a corrector if and only if the following hold for all n, k P Z : Y n ` ,k “ ` V n,k X n,k Y n,k and X n,k ` “ V n ` ,k ´ ` X n,k V n,k Y n,k ¯ . (4.1) Proof.
The cocycle property (e) is equivalent to B p x ´ e , x q ´ B p x ´ e , x q “ B p x ´ e ´ e , x ´ e q ´ B p x ´ e ´ e , x ´ e q for all x P Z . Together, the cocycle and the recovery properties (e) and (f) are equivalent to e B p x ´ e ,x q´ ω x “ ` e B p x ´ e ´ e ,x ´ e q´ ω x ´ e e ω x ´ e e ´ B p x ´ e ´ e ,x ´ e q and e B p x ´ e ,x q “ e ω x ´ ` e B p x ´ e ´ e ,x ´ e q´ ω x ´ e e ω x ´ e e ´ B p x ´ e ´ e ,x ´ e q ¯ holding for all x P Z . Plug in x “ p n ` , k q in the first equation and x “ p n, k q in the second one,then apply the definitions of V , X , and Y . (cid:3) We will see below that iterating the first equation in (4.1) gives Y n ` ,k “ ` n ÿ j “´8 n ź i “ j V i,k X i,k for all n, k P Z . (4.2)Compare with (2.12).Suppose now t ω x : x P Z u have an i.i.d. distribution P . Then t V n,k : n, k P Z u are also i.i.d.Suppose t X n, : n P Z u are independent of the V variables and have a stationary probabilitydistribution µ . Once the variables t V n,k : n, k P Z u and t X n, : n P Z u are known, the rest of thevariables t Y n,k : n P Z , k P Z ` u and t X n,k : n P Z , k P N u can be computed via (4.2) and the secondequation in (4.1). The resulting process t V n,k , X n,k , Y n,k : n P Z , k P Z ` u is clearly stationary undershifts in the n index.Let Φ p µ q be the distribution of t X n, : n P Z u . By the equivalence of stationary correctordistributions and stationary polymer measures with boundary, discussed in Section 2.4 (with here y “ Z e ), the problem of finding a stationary corrector distribution P with Ω -marginal P is thesame as finding a stationary fixed point for the map Φ. Indeed, this will ensure that the p V, X, Y q process is stationary under shifts in the k index. Inspecting the dependence on V in (4.1) and (4.2)one can quickly see that P will also satisfy the future-independence property.The purpose of the next section is to show that Φ is a mean-preserving contraction and hencehas at most one fixed point with a given mean. TATIONARY DIRECTED POLYMERS 9 Properties of the update map
The properties of the update map Φ that we prove here require fewer assumptions than our mainresults. Hence, this section has its own setting and notation, which we introduce next.We are given a stationary process X “ t X n : n P Z u and an i.i.d. sequence V “ t V n : n P Z u with X ą V ą P , with expectation E . Suppose E r| log X |s ă 8 and E r| log V |s ă 8 .Let I denote the shift-invariant σ -algebra of the process tp X n , V n q : n P Z u and assume that E r log X | I s ą E r log V s P -almost surely.(5.1) Lemma 5.1.
Suppose that Y “ t Y n : n P Z u satisfies the recursion Y n ` “ ` V n X ´ n Y n , P -almost surely and for all n P Z . (5.2) Then n ´ t ă Y ă 8u log Y n Ñ almost surely.Proof. Let a P R and ε ą U k “ log V k ´ log X k . Define F ε p a q “ a and F εn ` p a q “ log p ` exp t F εn p a q ` U n ´ E r U | I s ` ε uq for n P Z ` . An induction argument shows that for any n P N , we have F εn p a q “ log ´ ` n ´ ÿ m “ exp ! n ´ ÿ k “ m p U k ´ E r U | I s ` ε q ) ` exp ! a ` n ´ ÿ k “ p U k ´ E r U | I s ` ε q )¯ . As usual, we take an empty sum to be zero. The ergodic theorem implies then that F εn p a q “ log p ` e nε ` o p n q q and therefore n ´ F εn p a q Ñ ε P -almost surely, for any a P R .Another induction (using the fact that E r U | ¯ I s ď
0) shows that on the event 0 ă Y ă 8 , wehave 0 ď log Y n ď F εn p log Y q for all n P N . The claim of the lemma follows. (cid:3) Lemma 5.2.
The process Y “ t Y n : n P Z u given by Y n “ ` n ´ ÿ m “´8 n ´ ź k “ m V k X k (5.3) is the unique stationary and almost surely finite solution to (5.2) . For any other process Y forwhich (5.2) holds P -almost surely for all n P Z , Y must satisfy P lim j Ñ´8 | Y j | Ñ 8 ( ą andthen either Y is stationary and P p| Y | “ 8q ą or Y is not stationary.Proof. Suppose Y is given by (5.3). By the ergodic theorem and (5.1)lim m Ñ´8 | m | n ´ ÿ k “ m p log V k ´ log X k q “ E r log V ´ log X | I s ă ă Y n ă 8 almost surely. It is also clear that Y is stationary and that (5.3) implies (5.2).Conversely, let Y satisfy (5.2) P -almost surely and for all n P Z . Iterating (5.2) implies thatwhenever j ă n ´
1, we must have Y n “ ` n ´ ÿ m “ j ` n ´ ź k “ m V k X k ` Y j n ´ ź k “ j V k X k . (5.5)Suppose now that P ! lim j Ñ´8 | j | ´ log | Y j | ą ) “ , (5.6) where we take the convention that log . Then this and (5.4) imply that almost surelylim j Ñ´8 | Y j | ś n ´ k “ j V k X k “
0. In this case, taking j Ñ ´8 in (5.5) along a subsequence that realizesthis liminf implies Y is given by (5.3) almost surely and for all n P Z .If, alternatively, the probability in (5.6) is positive, then with positive probability | Y j | Ñ 8 as j Ñ ´8 . If we furthermore assume that Y is stationary, then the ergodic theorem implies that | Y | “ 8 on the event t| Y j | Ñ 8u . To see this last claim note that for any c ą | k | ´ ř i “ k t| Y k | ě c u Ñ P p| Y | ě c | I q almost surely. Consequently, P | Y | “ 8 , | Y k | Ñ 8 ( “ E “ t| Y | “ 8u t| Y k | Ñ 8u ‰ “ E “ P p| Y | “ 8 | I q t| Y k | Ñ 8u ‰ “ . (cid:3) Given the setting at the beginning of the section, define Y “ t Y n : n P Z u by (5.3). By Lemma5.2, Y satisfies (5.2) and 1 ă Y ă 8 almost surely. Define the stationary process X n “ V n ` p ` V ´ n X n Y ´ n q “ V n ` Y n ` X n V n Y n P p , , n P Z . (5.7)An induction argument shows that for n P N , we have Y V n ź k “ X k “ Y n ` V n ` n ź k “ X k . (5.8)Lemma 5.1 implies that log Y n { n Ñ X ą log V almost surely. Hence,the ergodic theorem implies log X is integrable and E r log X | I s “ E r log X | I s . (5.9) Lemma 5.3.
Suppose tp X n , V n , Y n , X n q : n P Z u and tp X n , V n , Y n , X n q : n P Z u both satisfyequations (5.3) and (5.7) . Note that both families share the same V variables. Suppose also that X n ď X n for all n P Z . Then Y n ě Y n and X n ď X n for all n P Z .Proof. It follows immediately that if X n ď X n for all n P Z , then Y n “ ` n ´ ÿ m “´8 n ´ ź k “ m V k X k ě ` n ´ ÿ m “´8 n ´ ź k “ m V k X k “ Y n . Then one has X n { Y n ď X n { Y n for all n P Z . It follows that X n “ V n ` ´ ` X n V n Y n ¯ ď V n ` ´ ` X n V n Y n ¯ “ X n . (cid:3) Now we define the update operator
Φ : M p R Z q Ñ M p R Z q , where M p X q is the set of proba-bility measures on X . Let Ω A “ Ω W “ R Z . Let A “ p A n q n P Z and p A , W q “ p A n , W n q n P Z be thenatural coordinate projections on Ω A and Ω A ˆ Ω W , respectively. Let p A , A q “ p A n , A n q n P Z and p A , A , W q “ p A n , A n , W n q n P Z be the natural coordinate projections on Ω A ˆ Ω A and Ω A ˆ Ω A ˆ Ω W ,respectively. Equip all these spaces with the product topologies, Borel σ -algebras, and naturalshifts. Let I and I be the σ -algebras of shift-invariant Borel subsets of Ω A and Ω A ˆ Ω W , respec-tively. Note that if we view Ω A as embedded in Ω A ˆ Ω W and abuse notation by identifying I and I ˆ Ω W , then I Ă I .For i P t , , u , given numbers p A in q n P Z and p W n q n P Z define X in “ e A in and V n “ e W n then Y in and X in via (5.3) and (5.7), n P Z . Let A in “ log X in .To define Φ we need a probability measure Γ on R such that ş | s | Γ p ds q ă 8 . Let M “ ş s Γ p ds q .Given such a Γ, the mapping Φ “ Φ Γ sends µ P M p Ω A q to the distribution Φ p µ q P M pr´8 , Z q of p A n q n P Z induced by P “ µ b Γ b Z P M p Ω A ˆ Ω W q .We are interested in shift-invariant fixed points of Φ. To have Φ p µ q P M p Ω A q we need to have | A n | ă 8 , P -almost surely. This is guaranteed if µ is satisfies E µ r| A |s ă 8 and µ -almost surely TATIONARY DIRECTED POLYMERS 11 E µ r A | I s ą E r W s “ M . Indeed, as observed above, I Ă I and hence the stochastic process p X , V q “ tp X n , V n q : n P Z u falls in the setting at the beginning of the section. Then by (5.7) wehave | A n | ă 8 almost surely.Given two stationary probability measures µ , µ on Ω A , let M p µ , µ q denote all stationaryprobability measures on Ω A ˆ Ω A , with marginals µ and µ . Recall the definition of the ¯ ρ distance:¯ ρ p µ , µ q “ inf λ P M p µ ,µ q E λ “ | A ´ A | ‰ . (5.10)When µ and µ are ergodic, the infimum may be taken over ergodic measures λ , by the ergodicdecomposition theorem. It is shown in [15, Theorem 8.3.1] that the ¯ ρ distance is a metric and theinfimum is achieved. The following is a positive-temperature analogue of an argument originallydue to Chang [4]. Note that the technical assumption P p V ą c q ą c , required in [4], isnot needed in positive temperature. Proposition 5.4.
Let µ and µ be two ergodic probability measures on Ω A . Assume E µ i r| A |s ă 8 and E µ i r A s ą M , i P t , u . Then ¯ ρ p Φ p µ q , Φ p µ qq ď ¯ ρ p µ , µ q . If in addition µ ‰ µ but E µ r A s “ E µ r A s , then ¯ ρ p Φ p µ q , Φ p µ qq ă ¯ ρ p µ , µ q .Proof. Fix an ergodic λ P M p µ , µ q and let P “ λ b Γ b Z P M p Ω A ˆ Ω A ˆ Ω W q . Being a productof an ergodic measure and a product measure, P is also ergodic. Let A n “ A n _ A n . Then E r A s “ E λ r A s ě E λ r A s “ E µ r A s ą M . Thus, the setting at the beginning of the section applies to X “ t X n “ e A n : n P Z u and V .Construct A “ t A n : n P Z u as was done above for A i , i P t , , u . Lemma 5.3 implies that P -almost surely A ě A _ A . Hence P -almost surely and for all n P Z | A ´ A | “ A _ A ´ A ´ A “ A ´ A ´ A and | A ´ A | “ A _ A ´ A ´ A ď A ´ A ´ A . (5.11)By (5.9) we have E r A i s “ E r log X i s “ E r log X i s “ E r A i s “ E λ r A i s for i P t , , u . This and(5.11) give E r| A ´ A |s ď E r A ´ A ´ A s “ E λ r A ´ A ´ A s “ E λ r| A ´ A |s . The left hand side is greater than ¯ ρ p Φ p µ q , Φ p µ qq . The first claim now follows by taking theinfimum over λ on the right hand side.Turning to the second claim, suppose that E µ r A s “ E µ r A s and that µ ‰ µ . Let λ P M p µ , µ q be an ergodic minimizer of (5.10). Then there exists an integer n ą λ A ă A , A m ď A m , ď m ă n, A n ą A n ( ą . Under the event in the above probability, we have ´ n ÿ k “ m A k ¯ _ ´ n ÿ k “ m A k ¯ ă n ÿ k “ m A k whenever 1 ď m ă n . This is equivalent to ´ n ź k “ m X k ¯ _ ´ n ź k “ m X k ¯ ă n ź k “ m X k . (5.12)We also have X k _ X k ď X k for all k P Z . Then the representation (5.3) of Y i , i P t , , u , gives Y n ` ă Y n ` ^ Y n ` and from (5.7), it follows that for i P t , u X in ` “ V n ` ´ ` X in ` V n ` Y in ` ¯ ă V n ` ´ ` X n ` V n ` Y n ` ¯ “ X n ` . Thus, λ ` A n ` _ A n ` ă A n ` ˘ ą . In particular,¯ ρ p Φ p µ q , Φ p µ qq ď E λ r| A ´ A |s “ E λ r| A n ` ´ A n ` |s“ E λ r A n ` _ A n ` ´ A n ` ´ A n ` să E λ r A n ` ´ A n ` ´ A n ` s “ E λ r| A ´ A |s “ ¯ ρ p µ , µ q . (cid:3) Let M αe p Ω A q be the set of ergodic probability measures µ P M p Ω A q with marginal mean E µ r A s “ α . The following is an immediate consequence of the previous proposition. Corollary 5.5.
For each α ą M there exists at most one µ P M αe p Ω A q with Φ p µ q “ µ . Lemma 5.6.
Suppose µ P M p Ω A q is stationary, Φ p µ q “ µ , and for some constant α ą M wehave lim n Ñ8 n n ´ ÿ m “ A m “ α, µ -almost surely. (5.13) Then µ P M αe p Ω A q .Proof. By the ergodic decomposition theorem, there exists Q µ P M ` M αe p Ω A q ˘ with µ “ ż M αe p Ω A q ν Q µ p dν q . Let φ be a bounded measurable function on Ω A . Then E Φ p µ q r φ p A qs “ E µ E Γ b Z r φ p A qs “ ż M αe p Ω A q E ν E Γ b Z r φ p A qs Q µ p dν q“ ż M αe p Ω A q E Φ p ν q r φ p A qs Q µ p dν q . This is equivalent to Φ p µ q “ ş Φ p ν q Q µ p dν q . Since Φ p µ q “ µ , uniqueness in the ergodic decomposi-tion theorem implies that Q µ ˝ Φ ´ “ Q µ . In particular, Φ p ν q P M αe p Ω A q for Q µ -almost every ν .Also, for any k P N ż M αe p Ω A q ¯ ρ p ν, Φ p ν qq Q µ p dν q “ ż M αe p Ω A q ¯ ρ p Φ k p ν q , Φ k ` p ν qq Q µ p dν q . The inequality in Proposition 5.4 then implies that Q µ ´! ν P M αe p Ω A q : ¯ ρ p ν, Φ p ν qq “ ¯ ρ p Φ k p ν q , Φ k ` p ν qq @ k P N )¯ “ . By the second part of Proposition 5.4 it must be the case that ν “ Φ p ν q for Q µ -almost every ν .Corollary 5.5 then implies that Q µ is a Dirac mass and so µ P M αe p Ω A q . (cid:3) We close this section with a proof of the stationarity mentioned at the end of Section 2.5.
Lemma 5.7.
Fix ρ ą θ ą . Assume t V n : n P Z u are i.i.d. such that { V is gamma-distributedwith scale parameter and shape parameter ρ . Assume X n are i.i.d. such that { X is gamma-distributed with scale parameter and shape parameter θ . Assume the two families of randomvariables are independent. Then t X n : n P Z u , defined by (5.7) , has the same distribution as t X n : n P Z u .Proof. Let Y be independent of t X n , V n : n P Z ` u with 1 { Y being gamma-distributed with scaleparameter 1 and shape parameter ρ ´ θ . Define t Y n : n P N u and t X n : n P Z ` u inductively using Y n ` “ ` V n X ´ n Y n and X n “ V n ` p ` V ´ n X n { Y n q . TATIONARY DIRECTED POLYMERS 13
The Burke property [27, Theorem 3.3] tells us that tp Y m ` n , X m ` n , X m ` n , V m ` n q : n P Z ` u has thesame distribution for all m P Z ` and that t X n : n P Z ` u has the same distribution as t X n : n P Z ` u .Using Kolmogorov’s extension theorem we can extend the above random variables to a family tp Y n , X n , X n , V n q : n P Z u . In particular, the distribution of X is the same as that of the process X and Y and X satisfy the above induction for all n P Z .By the uniqueness in Lemma 5.2, Y must equal the process Y defined by (5.3), which thenimplies that X is the same as X defined by (5.7). The claim now follows because we alreadyestablished that X has the same distribution as X . (cid:3) Proof of Theorems 3.1 and 3.5
Fix an i.i.d. probability measure P on p Ω , F q with L weights. Let P be a stationary future-independent L corrector distribution with Ω -marginal P . Let X n “ e B p ne , p n ` q e q , Y n “ e B p ne ,ne ` e q´ ω ne ` e , and V n “ e ω ne ` e , n P Z . Future independence implies the two processes X and V are independent of each other. Let I be the invariant σ -algebra for the process p X , V q .Let µ be the distribution of A “ t log X n : n P Z u . Let M “ E r ω s “ E r log V s . Recovery(f) implies that B p , e q ą ω e and hence E r log A | I s “ E r B p , e q | I s ą E r ω s . In particular, α “ m P ¨ e “ E r B p , e qs ą M . Lemma 6.1.
There exists a Borel-measurable map F : R Z ˆ R Z ˆ N Ñ R Z ˆ Z such that P -almostsurely, t B p x, y, ω q : x, y P Z ˆ Z ` u“ F ` t B p ne , p n ` q e q : n P Z u , t ω ne ` ke : n P Z , k P N u ˘ . Proof.
Lemma 4.1 implies that Y satisfies (5.2). Since it is a stationary almost surely finiteprocess, Lemma 5.2 implies that Y has the representation (5.3). Then the second equation in(4.1) says that e B p ne ` e , p n ` q e ` e q is equal to X n , defined by (5.7). This argument shows thatthe process t B p ne , ne ` e q , B p ne ` e , p n ` q e ` e q : n P Z u is a measurable function of t B p ne , p n ` q e q : n P Z u and t ω ne ` e : n P Z u .Since P is stationary, we have that A “ t log X n : n P Z u has the same distribution µ as A . In other words, Φ p µ q “ µ . This lets us repeat the above procedure inductively to get that t B p x, x ` e i q : x P Z ˆ Z ` , i “ , u is a measurable function of t B p ne , p n ` q e q : n P Z u and t ω ne ` ke : n P Z , k P N u . We also have B p x ` e i , x q “ ´ B p x, x ` e i q , P -almost surely. Then thecocycle property (e) implies that for x, y P Z ˆ Z ` , B p x, y q is the sum of B p x k , x k ` q along anypath with steps t˘ e , ˘ e u from x to y . The claim of the lemma follows. (cid:3) Corollary 6.2. If P is a stationary future-independent L corrector distribution with Ω -marginal P and the distributions of t B p ne , p n ` q e q : n P Z u under P and P match, then P “ P .Proof. Lemma 6.1 implies that the distributions of t B p x, y q : x, y P Z ˆ Z ` u under P and P match.Then stationarity of the two probability measures implies P “ P . (cid:3) Proof of Theorem 3.1.
As was mentioned in the proof of Lemma 6.1, µ is a fixed point of Φ.Corollary 5.5 says that there exists at most one ergodic such µ . Corollary 6.2 thus implies thatthere exists at most one T e -ergodic P . Switching e and e around in the definitions of X and Y we get the same result for the T e shift. (cid:3) Proof of Theorem 3.5.
Theorem 4.4 and Lemma 4.5(c) in [18] imply that n ´ B p , ne q convergesalmost surely to m P ¨ e . Since B p , ne q “ ř n ´ m “ A m we have that (5.13) holds and Lemma 5.6says that µ is ergodic. Corollary 6.2 says P is determined by µ b Γ bp Z ˆ Z ` q . Since this is a productof an ergodic measure and a product measure, it is ergodic. Ergodicity of P under the T e shiftfollows. A symmetric argument gives the ergodicity under the T e shift. (cid:3) References [1]
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J. Phys. A Christopher Janjigian, University of Utah, Mathematics Department, 155 S 1400 E, Salt Lake City,UT 84112, USA.
E-mail address : [email protected] URL : Firas Rassoul-Agha, University of Utah, Mathematics Department, 155S 1400E, Salt Lake City,UT 84112, USA.
E-mail address : [email protected] URL ::