Properties of the limit shape for some last passage growth models in random environments
aa r X i v : . [ m a t h . P R ] S e p PROPERTIES OF THE LIMIT SHAPE FOR SOME LAST PASSAGEGROWTH MODELS IN RANDOM ENVIRONMENTS
HAO LIN AND TIMO SEPP ¨AL ¨AINEN
Abstract.
We study directed last passage percolation on the planar square lattice whoseweights have general distributions, or equivalently, queues in series with general service distri-butions. Each row of the last passage model has its own randomly chosen weight distribution.We investigate the limiting time constant close to the boundary of the quadrant. Close to the y -axis, where the number of random distributions averaged over stays large, the limiting timeconstant takes the same universal form as in the homogeneous model. But close to the x -axiswe see the effect of the tail of the distribution of the random environment. Introduction
This paper studies the limit shapes of some last passage percolation models in random en-vironments, specifically, the corner growth model and two Bernoulli models with different rulesfor admissible paths.We introduce the corner growth model through its queueing interpretation. Consider servicestations in series, labeled 0 , , , . . . , ℓ , each with unbounded waiting room and first-in first-out(FIFO) service discipline. Initially customers 0 , , , . . . , k are queued up at server 0. At time t = 0 customer 0 begins service with server 0. Each customer moves through the system ofservers in order, joining the queue at server j + 1 as soon as service with server j is complete.After customer i departs server j , server j starts serving customer i + 1 immediately if i + 1has been waiting in the queue, or then waits for customer i + 1 to arrive from station j − X ( i, j ) be the service time that customer i needs at station j , and T ( k, ℓ ) the time when customer k completes service with server ℓ .Asymptotics for T ( k, ℓ ) as k and ℓ get large have been investigated a great deal in the pasttwo decades. A seminal paper by Glynn-Whitt [6] studied the case of i.i.d. { X ( i, j ) } . They tookadvantage of the connection with directed last-passage percolation given by the identity(1.1) T ( k, ℓ ) = max π X ( i,j ) ∈ π X ( i, j ) . The maximum is taken over non-decreasing nearest-neighbor lattice paths π ⊆ Z from (0 , k, ℓ ) that are of the form π = { (0 ,
0) = ( x , y ) , ( x , y ) , . . . , ( x k + ℓ , y k + ℓ ) = ( k, ℓ ) } where( x i , y i ) − ( x i − , y i − ) = (1 ,
0) or (0 , corner growth model . Date : November 12, 2018.2000
Mathematics Subject Classification.
Key words and phrases. corner growth model, random environment, limit shape, last passage percolation, queuesin series.T. Sepp¨al¨ainen was partially supported by National Science Foundation grants DMS-0701091 and DMS-1003651,and by the Wisconsin Alumni Research Foundation.
Next we add a random environment to this queueing model. The environment is a sequence { F j : j ∈ Z + } of probability distributions, generated by a probability measure-valued ergodicor i.i.d. process with distribution P . Given the sequence { F j } , we assume that the variables { X ( i, j ) } are independent and X ( i, j ) has distribution F j . In the queueing picture this meansthat the service times { X ( i, j ) : i ∈ Z + } at service station j have common distribution F j , andat the outset the distributions { F j : j ∈ Z + } themselves are chosen randomly according to somegiven law P . Obviously the labels “customer” and “server” are interchangeable because we canswitch around the roles of the indices i and j .The asymptotic regime we consider for T ( k, ℓ ) is the hydrodynamic one where k and ℓ areboth of order n and n is taken to ∞ . Under some moment assumptions standard subadditiveconsiderations and approximations imply the existence of the deterministic limitΨ( x, y ) = lim n →∞ n − T ( ⌊ nx ⌋ , ⌊ ny ⌋ ) for all ( x, y ) ∈ R .Only in the case where the distributions F j are exponential or geometric has it been possibleto describe explicitly the limit Ψ. This is the case of · /M/ { X ( i, j ) } the limit Ψ( x, y ) = ( √ x + √ y ) was first derived by Rost [17] in a seminal paper onhydrodynamic limits of asymmetric exclusion processes. The random environment model withexponential F j ’s was studied in [1, 12, 20].Let us now set aside the queueing motivation and consider the last-passage model on thefirst quadrant Z of the planar integer lattice, defined by the nondecreasing lattice paths andthe random weights { X ( i, j ) } . For the queueing application it is natural to assume the weightsnonnegative, but in the general last-passage situation there is no reason to restrict to nonnegativeweights.The ideal limit shape result would have some degree of universality, that is, apply to a broadclass of distributions. Such results have been obtained only close to the boundary: in [14] Martinshowed that in the i.i.d. case, under suitable moment hypotheses and as α ց , α ) = µ + 2 σ √ α + o ( √ α ) , where µ and σ are the common mean and variance of the weights X ( i, j ). Also, the o ( √ α )term in the statement means that lim α ց α − / (cid:2) Ψ(1 , α ) − µ − σ √ α (cid:3) = 0 . In the i.i.d. case Ψis symmetric so the same holds for Ψ( α, { X ( i, j ) } is not invariant undertransposition. So we must ask the question separately for Ψ(1 , α ) and Ψ( α, α, α,
1) = µ + 2 σ √ α + o ( √ α ) as α ց
0, where now µ is the mean as before but σ is theaverage of the “quenched” variance. That is, if we let µ = R x dF ( x ) and σ = R ( x − µ ) dF ( x )denote the mean and variance of the random distribution F , and E expectation under P , then µ = E ( µ ) and σ = E ( σ ).The case Ψ(1 , α ) does not possess a clean result such as the one above. Even though we arestudying the deterministic limit obtained after n has been taken to infinity, we see an effectfrom the tail of the distribution of the quenched mean µ . We illustrate this with the case ofexponential { F j } . Now the number nα of distributions F j is small compared to the number n ofweights X ( i, j ) in each row, hence the fluctuations among the F j ’s become prominent. The effectcomes in two forms: first, the leading term is no longer the averaged mean µ but the maximal AST-PASSAGE SHAPE 3 mean. Second, if large values among the row means µ j are rare, the order of the α -dependentcorrection is smaller than the √ α seen above and this order of magnitude depends on the tailof the distribution of µ . As an exponent characterizing this tail changes, we can see a phasetransition of sorts in the power of α , with a logarithmic correction at the transition point.For general distributions we derive bounds on Ψ(1 , α ) that indicate that in the case of finitelymany distributions the correction is of order √ α .As auxiliary results we need bounds on the limits for last-passage models with Bernoulliweights under a random environment. However, with Bernoulli weights the standard cornergrowth model is not one of the explicitly solvable cases. The model with Bernoulli weights doesbecome solvable when the path geometry is altered suitably. The model we take up is the onewhere the paths are weakly increasing in one coordinate but strictly in the other. There aretwo cases, depending on which coordinate is required to increase strictly. If we require the x -coordinate to increase strictly then an admissible path { ( x , y ) , ( x , y ) , . . . , ( x m , y m ) } satisfies(1.3) x i +1 − x i = 1 and y ≤ y ≤ · · · ≤ y m .The other case interchanges x and y . These cases have to be addressed separately because therandom environment attached to rows makes the model asymmetric. The sum of these twolast-passage values gives a bound for the case where neither coordinate is required to increasestrictly in each step.We derive the exact limit constants for Bernoulli models with both types of strict/weak paths.For one of them this has been done before by Gravner, Tracy and Widom [9]. Their proof utilizesthe fact that the distribution of T ( k, ℓ ) is a symmetric function of the environment (at least forthe particular Bernoulli case they study). Our proof is completely different. It is based on theidea in [19] where the limit for the homogeneous case was derived: the last-passage model iscoupled with a particle system whose invariant distributions can be written down explicitly, andthen through some convex analysis the speed of a tagged particle yields the explicit limit of thelast-passage model. Further remarks on the literature.
The present paper does not address questions of fluctua-tions, but let us mention some highlights from the literature. For the last-passage model withi.i.d. exponential or geometric weights, the distributional limit with fluctuations of order n / and limit given by the Tracy-Widom GUE distribution was proved by Johansson [10]. As forthe shape, universality has been achieved only close to the boundary, by Baik-Suidan [2] andBodineau-Martin [3].Fluctuations of the Bernoulli model with strict/weak paths and homogeneous weights werederived first in [11] and later also in [7]. For the model in a random environment fluctuationlimits appear in [9, 8].On the lattice Z we can imagine three types of nondecreasing paths: (i) weak-weak: bothcoordinates required to increase weakly, the type used in (1.1); (ii) strict-weak: one coordinateincreases strictly, as above in (1.3); and (iii) strict-strict: both coordinates increase strictly soan admissible path { ( x , y ) , ( x , y ) , . . . , ( x m , y m ) } satisfies x < · · · < x m and y < · · · < y m .As mentioned, with Bernoulli weights the strict-weak case is solvable but the weak-weak caseappears harder. The third case, strict-strict, is also solvable with Bernoulli weights. The shapewas derived in [18] and recent work on this model appears in [5]. Organization of the paper.
The main results on the shape close to the boundary are in Section2 and the results for Bernoulli models in Section 3. Section 4 sketches the proof of the existenceof the limiting shape, a result we basically take for granted. The main proofs follow: in Section5 for Theorem 2.2 on Ψ( α, , α ), and in Section 7 forTheorem 2.4 for the exponential model. HAO LIN AND TIMO SEPP ¨AL¨AINEN
Some frequently used notation.
We writeess sup P f = inf { s ∈ R : P ( f > s ) = 0 } for the essential supremum of a function f under a measure P . Z + = { , , , . . . } , N = { , , , . . . } , and R + = [0 , ∞ ). I ( A ) is the indicator function of event A .2. Main results
First a precise definition of the last-passage model in a random environment. Let P be astationary, ergodic probability measure on the space M ( R ) Z + of sequences of Borel probabilitydistributions on R . E denotes expectation under P . For some of the main results P will beassumed to be an i.i.d. product measure. A realization of the distribution-valued process under P is denoted by { F j } j ∈ Z + . This is the environment. Given { F j } , the weights { X ( z ) : z ∈ Z } areindependent real-valued random variables with marginal distributions X ( i, j ) ∼ F j for ( i, j ) ∈ Z . Let (Ω , F , P ) be the probability space on which all variables { F j , X ( i, j ) } are defined, anddenote expectation under P by E .A (weakly) nondecreasing path is a sequence of points z = ( x , y ) , z = ( x , y ) , . . . , z m =( x m , y m ) in Z that satisfy x ≤ x ≤ · · · ≤ x m , y ≤ y ≤ · · · ≤ y m , and | x i +1 − x i | + | y i +1 − y i | = 1. For z , z ∈ Z with z ≤ z (coordinatewise ordering), let Π( z , z ) be the set ofnondecreasing paths from z to z . Whether the endpoints z and z are included in the pathmakes no difference to the limit results below. The last-passage time T ( z , z ) from z to z isdefined by T ( z , z ) = max π ∈ Π( z ,z ) X z ∈ π X ( z ) . When z = 0 abbreviate Π( z ) = Π(0 , z ) and T ( z ) = T (0 , z ).Put these three assumptions on the model:(2.1) E | X ( z ) | < ∞ , (2.2) Z ∞ n − E ( F ( x )) o / dx < ∞ , and(2.3) Z ∞ ess sup P (1 − F ( x )) dx < ∞ . We begin with this by now standard result that defines our object of study, namely the functionΨ. The proof is briefly commented on in Section 4.
Proposition 2.1.
Assume P is ergodic and satisfies (2.1) , (2.2) and (2.3) . Then for all ( x, y ) ∈ (0 , ∞ ) the last passage time constant (2.4) Ψ( x, y ) = lim n →∞ n T ( ⌊ nx ⌋ , ⌊ ny ⌋ ) exists as a limit both P -almost surely and in L ( P ) . Furthermore, Ψ( x, y ) is a homogeneous,concave and continuous function on (0 , ∞ ) . Assumption (2.2) is also used for the constant distribution case, see (2.5) in [14]. Somefurther control along the lines of assumption (2.3) is required for our case. For example, suppose1 − F j ( x ) = e − ξ j x for random ξ j ∈ (0 , ∞ ). Then (2.3) holds iff ess inf P ( ξ ) >
0. If the distributionof ξ is not bounded away from zero, n − T ( n, n ) → ∞ because we can simply collect all theweights from the row with minimal ξ j among { ξ , . . . , ξ n } . However, assumption (2.2) can besatisfied without bounding ξ away from zero. AST-PASSAGE SHAPE 5
Now we turn to the main results of the paper on the form of the limit shape at the boundary.As explained in the introduction, for Ψ( α,
1) we find a universal form as α ց
0. In addition tothe earlier assumptions, we need similar control of the left tail of the distributions:(2.5) Z −∞ (cid:0) E [ F ( x )] (cid:1) / dx < ∞ and(2.6) Z −∞ ess sup P F ( x ) dx < ∞ . Let us point out that (2.2) and (2.5) together guarantee E | X ( z ) | < ∞ . Let µ j = µ ( F j ) and σ j = σ ( F j ) denote the mean and variance of distribution F j . These are random variables under P with expectations µ = E ( µ ) and σ = E ( σ ). Theorem 2.2.
Assume the process { F j } is i.i.d. under P , and satisfies tail assumptions (2.2) , (2.3) , (2.5) and (2.6) . Then, as α ↓ , Ψ( α,
1) = µ + 2 σ √ α + o ( √ α ) . Assumptions (2.2) and (2.5) are direct counterparts of what was used for Theorem 2.4 in[14]. Assumptions (2.3) and (2.6) are additional assumptions needed for handling the randomenvironment. These assumptions are used to control estimates that come from bounding limitsof Bernoulli models.We turn to the case Ψ(1 , α ). The results will be qualitatively different from Theorem 2.2.The leading term will be the essential supremum of the mean instead of the averaged mean andwe will see different orders for the first α -dependent correction term.First a general result for which we restrict ourselves to the case of finitely many distributions,but we can relax the i.i.d. assumption of the random distributions. Theorem 2.3.
Assume the process { F j } of probability distributions is stationary, ergodic, andhas a state space of finitely many distributions H , . . . , H L each of which satisfies Martin’s [14] hypothesis (2.7) Z ∞ (1 − H ℓ ( x )) / dx + Z −∞ H ℓ ( x ) / dx < ∞ . Let µ ∗ = max ℓ µ ( H ℓ ) be the maximal mean of the H ℓ ’s. Then there exist constants < c
0. Then the essential supremum of therandom mean is µ ∗ = c − .An implicit description of the limit shape was derived in [20] by way of studying an exclusionprocess with random jump rates attached to particles. We recall the result here. One explicitshape is needed for the proof of Theorem 2.2 also, so this result will serve there too. HAO LIN AND TIMO SEPP ¨AL¨AINEN
Define first a critical value u ∗ = R [ c, ∞ ) cξ − c m ( dξ ) ∈ (0 , ∞ ]. For 0 ≤ u < u ∗ define a = a ( u )implicitly by u = Z [ c, ∞ ) aξ − a m ( dξ ) .a ( u ) is strictly increasing, strictly concave, continuously differentiable and one-to-one from 0 G ( x, y ) ≥ } and(2.10) Ψ G ( x, y ) = inf { t ≥ tg ( y/t ) ≥ x } . Qualitative properties of the limit shape depend on the tail of the distribution m at c +,and transitions occur where the integrals R [ c, ∞ ) ( ξ − c ) − m ( dξ ) and R [ c, ∞ ) ( ξ − c ) − m ( dξ ) blowup. (For details see [20].) These same regimes appear in our results below. For the case R [ c, ∞ ) ( ξ − c ) − m ( dξ ) = ∞ we make a precise assumption about the tail of the distribution ofthe random rate:(2.11) ∃ ν ∈ [ − , , κ > ξ ց c m [ c, ξ )( ξ − c ) ν +1 = κ. The value ν = − c has probability m { c } = κ >
0. Values ν < − Theorem 2.4.
For the model with exponential distributions with i.i.d. random rates the limit Ψ G has these asymptotics close to the x -axis. Case 1: R [ c, ∞ ) ( ξ − c ) − m ( dξ ) < ∞ . Then there exists α > such that (2.12) Ψ G (1 , α ) = c − + α Z [ c, ∞ ) ξ − c m ( dξ ) for α ∈ [0 , α ] . Case 2: (2.11) holds so that, in particular R [ c, ∞ ) ( ξ − c ) − m ( dξ ) = ∞ . Then as α ց ,if ν ∈ (0 , then Ψ G (1 , α ) = c − + α Z [ c, ∞ ) ξ − c m ( dξ ) + o ( α ) ;(2.13) if ν = 0 then Ψ G (1 , α ) = c − − κα log α + o ( α log α ) ;(2.14) if ν ∈ [ − , then Ψ G (1 , α ) = c − + Bα − ν + o ( α − ν ) . (2.15)In statement (2.15) above B = B ( c, κ, ν ) is a constant whose explicit definition is in equation(7.7) in the proof section below. The extreme case ν = − n Ψ(1 , α ) represents the time when customer n departsfrom server ⌊ nα ⌋ (rigorously speaking in the n → ∞ limit), when initially all customers arequeued up at server 0. When u ∗ < ∞ (Case 1 and subcase ν > j (whose service rate is ξ j ) is geometric withmean c/ ( ξ j − c ) and the departure process from each queue has rate c . A customer arriving at aqueue with this geometric number of customers present spends on average time 1 / ( ξ j − c ) at that AST-PASSAGE SHAPE 7 queue. In Case 1 this picture is precise enough so that equation (2.12) can be naively understoodin these terms: a single customer travels through ⌊ nα ⌋ servers in time nα R [ c, ∞ ) 1 ξ − c m ( dξ ). Afterthis it takes another n/c time to see n customers go through server ⌊ nα ⌋ . Together these termsmake up the right-hand side of (2.12). This argument requires a high density of customersbecause once the customer density drops below u ∗ the system chooses an equilibrium with flowrate below c . (Again, for details we refer to [1].)This point can be detected in the proofs for Case 2 where we find a parameter a that in somesense represents a flow rate and replaces c in (2.12) (see eqn. (7.2)). The formulas in Case 2 arethen obtained by estimating c − a .3. Bernoulli models with strict-weak paths in a random environment
This section looks at last-passage models with Bernoulli-distributed weights. The environmentis now an i.i.d. sequence { p j } j ∈ Z + of numbers p j ∈ [0 , P . Given { p j } , theweights { X ( i, j ) } are independent with marginal distributions P ( X ( i, j ) = 1) = p j = 1 − P ( X ( i, j ) = 0). We consider two last-passage times that differ by the type of admissible path:for z , z ∈ Z (3.1) T → ( z , z ) = max π ∈ Π → ( z ,z ) X z ∈ π X ( z ) and T ↑ ( z , z ) = max π ∈ Π ↑ ( z ,z ) X z ∈ π X ( z ) . In terms of coordinates denote the endpoints by z k = ( a k , b k ), k = 1 ,
2. Then admissiblepaths π ∈ Π → ( z , z ) are of the form π = { ( a , y ) , ( a + 1 , y ) , ( a + 2 , y ) , . . . , ( a , y a − a ) } with b ≤ y ≤ y ≤ · · · ≤ y a − a ≤ b , while paths π ∈ Π ↑ ( z , z ) are of the form π = { ( x , b ) , ( x , b + 1) , . . . , ( x b − b , b ) } with a ≤ x ≤ x ≤ · · · ≤ x b − b ≤ a . Thus paths inΠ → ( z , z ) increase strictly in the x -direction while those in Π ↑ ( z , z ) increase strictly in the y -direction. The last-passage times T → ( z , z ) and T ↑ ( z , z ) record the maximal weights of suchpaths in the lattice rectangle ([ a , a ] × [ b , b ]) ∩ Z . The following figures illustrate the twotypes of admissible paths when we take z = (0 ,
0) and z = (5 , ✲✻ ✉ ✉ ✉ ✉ ✉ ✉ ✲(cid:0)(cid:0)✒(cid:0)(cid:0)✒ ✲✂✂✂✂✂✂✂✍ ij (a) ✲✻ ✉ ✉ ✉✉ ✉✉ ✟✟✟✟✯(cid:0)(cid:0)✒✻✟✟✟✟✯✻ ij (b) Figure 1.
Admissible paths in Π → ( z , z ) and Π ↑ ( z , z ) HAO LIN AND TIMO SEPP ¨AL¨AINEN
As before we simplify notation with T → (0 , z ) = T → ( z ). The almost sure limits are denotedby(3.2) Ψ → ( x, y ) = lim n →∞ n T → ( ⌊ nx ⌋ , ⌊ ny ⌋ ) and Ψ ↑ ( x, y ) = lim n →∞ n T ↑ ( ⌊ nx ⌋ , ⌊ ny ⌋ )for ( x, y ) ∈ (0 , ∞ ) . The existence of the limits needs no further comment.The next theorem gives the explicit limits. (3.3) is the same as in [9, Thm. 1]. Inside the E ( · · · ) expectations below p is the random Bernoulli probability. Let b = ess sup P p denote themaximal probability. Theorem 3.1.
The limits in (3.2) are as follows for x, y ∈ (0 , ∞ ) . (3.3) Ψ → ( x, y ) = bx + y (1 − b ) E (cid:0) pb − p (cid:1) , x/y ≥ E (cid:0) p (1 − p )( b − p ) (cid:1) yz E (cid:0) − p ( z − p ) (cid:1) − y, E (cid:0) p − p (cid:1) < x/y < E (cid:0) p (1 − p )( b − p ) (cid:1) x, < x/y ≤ E (cid:0) p − p (cid:1) with z ∈ ( b, uniquely defined by the equation x/y = E h p (1 − p )( z − p ) i . (3.4) Ψ ↑ ( x, y ) = y − yz E (cid:0) − p ( z + p ) (cid:1) , < x/y < E (cid:0) − pp (cid:1) y, x/y ≥ E (cid:0) − pp (cid:1) with z ∈ (0 , ∞ ) uniquely defined by the equation x/y = E h p (1 − p )( z + p ) i . Our second result gives simplified bounds that are useful for the proof of the main resultTheorem 2.2. Let ¯ p = E ( p ) be the mean of the environment. Ψ( x, y ) is the limiting timeconstant with weakly increasing paths defined in Proposition 2.1. Theorem 3.2.
The following three inequalities hold for the Bernoulli model: (3.5) Ψ → ( x, y ) ≤ bx + 2 p ¯ p (1 − b ) xy, (3.6) Ψ ↑ ( x, y ) ≤ ¯ py + 2 p ¯ p (1 − ¯ p ) xy and (3.7) Ψ( x, y ) ≤ ¯ py + 4 p ¯ p (1 − ¯ p ) xy + bx. (3.7) follows from (3.5) and (3.6) because Ψ( x, y ) ≤ Ψ → ( x, y ) + Ψ ↑ ( x, y ). Another looseestimate we will use later following (3.7) is(3.8) Ψ( x, y ) ≤ ¯ py + 4 p ¯ p (1 − ¯ p ) xy + bx ≤ ( y + 4 √ xy ) √ ¯ p + bx. We prove the formulas and inequalities first for Ψ → and then for Ψ ↑ . For some parts of theproofs it is convenient to assume b <
1. Results for the case b = 1 follow by taking a limit. AST-PASSAGE SHAPE 9
Proof of (3.3) and (3.5) . We adapt the proof from [19] to the random environment situationand sketch the main points.Consider now the environment { p j } fixed, but the weights X ( i, j ) random. For integers0 ≤ s < t and a, k define an inverse to the last passage time asΓ(( a, s ) , k, t ) = min { l ∈ Z + : T → (( a + 1 , s + 1) , ( a + l, t )) ≥ k } . Note that Γ(( a, s ) , , t ) = 0 but Γ(( a, s ) , k, t ) > k >
0. Knowing the limits of the variablesΓ is the same as knowing Ψ → . By the homogeneity of Ψ → it is enough to find h ( x ) = Ψ → ( x, → , h is concave and nondecreasing. Let g be theinverse function of h on R + . Then g is convex and nondecreasing, and tg ( x/t ) = lim n →∞ n Γ((0 , , ⌊ nx ⌋ , ⌊ nt ⌋ ) . To find these functions we construct an exclusion-type process z ( t ) = { z k ( t ) : k ∈ Z } oflabeled, ordered particles z k ( t ) < z k +1 ( t ) that jump leftward on the lattice Z , in discretetime t ∈ Z + . Given an initial configuration { z i (0) } that satisfies z i − (0) ≤ z i (0) − i →−∞ | i | − z i (0) > − /b , the evolution is defined by(3.9) z k ( t ) = inf i : i ≤ k { z i (0) + Γ(( z i (0) , , k − i, t ) } , k ∈ Z , t ∈ N . It can be checked that z ( t ) is a well-defined Markov process, in particular that z k ( t ) > −∞ almost surely.Define the process { η i ( t ) } of interparticle distances by η i ( t ) = z i +1 ( t ) − z i ( t ) for i ∈ Z and t ∈ Z + . By Prop. 1 in [19] process { η i ( t ) } has a family of i.i.d. geometric invariant distributionsindexed by the mean u ∈ [1 , b − ) and defined by(3.10) P ( η i = n ) = u − (1 − u − ) n − , n ∈ N . Let x k ( t ) = z k ( t − − z k ( t ) ≥ k th particle from time t − t , and let q t = 1 − p t . From (6.5) in [19], in the stationary process(3.11) P ( x k ( t ) = x ) = ( (1 − up t ) q − t x = 0 p t (1 − up t ) q − t ( u − x ( uq t ) − x x = 1 , , , . . . We track the motion of particle z ( t ) in a stationary situation. The initial state is defined bysetting z (0) = 0 and by letting { η i (0) } be i.i.d. with common distribution (3.10). With k = 0,divide by t in (3.9) and take t → ∞ . Apply laws of large numbers inside the braces in (3.9),with some simple estimation to pass the limit through the infimum, to find the average speed ofthe tagged particle:(3.12) − lim t →∞ t z ( t ) = sup x ≥ { ux − g ( x ) } ≡ f ( u ) . The last equality defines the speed f as f = g + , the monotone conjugate of g . It is natural toset f ( u ) = 0 for u ∈ [0 , f ( b − ) = f (( b − ) − ), and f ( u ) = ∞ for u > b − . By [16, Thm. 12.4](3.13) g ( x ) = sup u ≥ { xu − g + ( u ) } = sup ≤ u ≤ /b { xu − f ( u ) } . Since z ( t ) is a sum of jumps x ( k ) with distribution (3.11) we have the second moment boundsup t ∈ N E [( t − z ( t )) ] < ∞ , and consequently the limit in (3.12) holds also in expectation. From this(3.14) f ( u ) = − lim t →∞ E [ t − z ( t )] = lim t →∞ E h t − t X k =1 x ( k ) i = E [ x (0)]= E ∞ X x =1 x ( u − x ( uq ) − x p (1 − up )(1 − p ) − = E h pu ( u − − up i . We find g ( x ) from (3.14) and (3.13):(3.15) g ( x ) = x/b − b − (1 − b ) E p ( b − p ) x ≥ b E (1 − p )( b − p ) − u E p (1 − p )(1 − u p ) E p − p < x < b E (1 − p )( b − p ) − x < x < E p − p where u ∈ (1 , b − ) is uniquely defined by the equation x + 1 = E (1 − p )(1 − u p ) − . From thiswe find the inverse function h ( x ) = g − ( x ) and then Ψ → ( x, y ) = yh ( x/y ). We omit these detailsand consider (3.3) proved.To prove (3.5) we return to the duality (3.13) and write(3.16) g ( x ) ≥ sup ≤ u< /b { xu − ˜ f ( u ) } for ˜ f ( u ) = u ( u − − ub ¯ p. ˜ f ′ ( u ) = x is solved by u ∗ = b − (cid:16) − q (1 − b )¯ pbx +¯ p (cid:17) . When x ≥ ¯ p − b , we have u ∗ ∈ [1 , b ), and then g ( x ) ≥ xu ∗ − ˜ f ( u ∗ ) = 1 b (cid:0)p (1 − b )¯ p − p bx + ¯ p (cid:1) . Consequently g − ( x ) ≤ b (cid:0) √ b x + p (1 − b )¯ p (cid:1) − ¯ pb = bx − ¯ p + 2 p (1 − b )¯ px. When x < ¯ p − b , the supremum in (3.16) is attained at u = 1, and in this case g − ( x ) ≤ x ≤ bx + 2 p (1 − b )¯ px. The bound (3.5) now follows from Ψ → ( x, y ) = yg − ( x/y ) . (cid:3) Proof of (3.4) and (3.6) . The scheme is the same, so we omit some more details. The inverse ofthe last-passage time is now definedΓ(( a, s ) , k, t ) = min { l ∈ Z + : T ↑ (( a, s + 1) , ( a + l, t )) ≥ k } . Vertical distance t − s allows for at most t − s marked points, so the above quantity must be setequal to ∞ for k > t − s . The particle process { z ( t ) : t ∈ Z + } is defined by the same formula (3.9)as before but it is qualitatively different. The particles still jump to the left, but the orderingrule is now z k ( t ) ≤ z k +1 ( t ) so particles are allowed to sit on top of each other. Well-definednessof the dynamics needs no further restrictions on admissible particle configurations because theminimum in (3.9) only considers i ∈ { k − t, . . . , k } so it is well-defined for all initial configurations { z i (0) : i ∈ Z } such that z i (0) ≤ z i +1 (0).The following can be checked. Under a fixed environment { p j } , the gap process { η i ( t ) = z i +1 ( t ) − z i ( t ) : i ∈ Z } has i.i.d. geometric invariant distributions P ( η k = n ) = ( u )( u u ) n , AST-PASSAGE SHAPE 11 n ∈ Z + , indexed by the mean u ∈ R + . In this stationary situation the successive jumps x k ( t ) = z k ( t − − z k ( t ) of a tagged particle have distribution P ( x k ( t ) = y ) = ( up t y = 0( uu +1 ) y p t up t y ≥ . From here the analysis proceeds the same way as for the other model. The speed function isdefined by f ( u ) = − lim n →∞ E (cid:2) n − z ( n ) (cid:3) = E [ x (0)] = u ( u + 1) E h p up i and then convex analysis takes over. We omit the remaining details of the proof of (3.4).To prove (3.6), note that g ( x ) = sup u ≥ { xu − f ( u ) } ≥ sup u ≥ n xu − ¯ pu ( u + 1)1 + u ¯ p o = ( p ( √ − x − √ − ¯ p ) ¯ p ≤ x ≤
10 0 ≤ x ≤ ¯ p. We used Jensen’s inequality and concavity of p p up . From this g − ( x ) ≤ ( ¯ p − ¯ px + 2 p ¯ p (1 − ¯ p ) x ≤ x ≤ − ¯ p ¯ p x > − ¯ p ¯ p and (3.6) follows. (cid:3) Proof of Proposition 2.1
We comment briefly on the proof of Proposition 2.1. Further details can be found in [13]. Theflow of arguments is standard. First one takes an integer point ( x, y ) ∈ Z and applies Liggett’sversion of the subadditive ergodic theorem to the process Z m,n = − T (( mx, my ) , ( nx, ny )), 0 ≤ m < n , to prove that Ψ( x, y ) exists and is finite. Then rational ( x, y ) and real ( x, y ) arehandled by approximations. Along the way regularity properties of Ψ are established and used:superadditivity, homogeneity, concavity and continuity.All this works easily for the Bernoulli case because last-passage times are uniformly boundedin terms of path length. Consequently we can assume that Proposition 2.1 has been proved forthe Bernoulli case. For the general case we check that for integer points ( x, y ) ∈ Z the momenthypotheses of the subadditive ergodic theorem [4, p. 358] follow from our assumptions (2.1),(2.2), and (2.3): E Z +0 , ≤ E | T ((0 , , ( x, y )) | ≤ E X ≤ i ≤ x, ≤ j ≤ y | X ( i, j ) | = ( x + 1)( y + 1) E | X (0 , | < ∞ . Next:1 n E Z ,n ≥ − n E max π ∈ Π( nx,ny ) X z ∈ π X ( z ) + = − n E max π ∈ Π( nx,ny ) X z ∈ π Z ∞ I ( X ( z ) > u ) du ≥ − n E Z ∞ max π ∈ Π( nx,ny ) X z ∈ π I ( X ( z ) > u ) du = − n Z ∞ E max π ∈ Π( nx,ny ) X z ∈ π I ( X ( z ) > u ) du ≥ − Z ∞ sup n n E max π ∈ Π( nx,ny ) X z ∈ π I ( X ( z ) > u ) du = − Z ∞ Ψ Ber [1 − F ( u )] ( x, y ) du ≥ − ( y + 4 √ xy ) Z ∞ p − E F ( u ) du − x Z ∞ (cid:0) − ess inf P F ( u ) (cid:1) du.I ( A ) is the indicator function of event A . Ψ Ber [1 − F ( u )] ( x, y ) is the limiting time constant forthe Bernoulli model where the weights have distributions P ( X ( i, j ) = 1) = 1 − F j ( u ) = 1 − P ( X ( i, j ) = 0). On the last line above we used the Bernoulli estimate (3.8). By assumptions(2.2) and (2.3), E Z ,n ≥ nγ for a constant γ > −∞ . These estimates justify the application ofthe subadditive ergodic theorem. We omit the remaining details and consider Proposition 2.1proved. 5. Proof of Theorem 2.2
For the first lemma, let { F j } and { G j } be ergodic sequences of distributions defined on acommon probability space under probability measure P . In a later step of the proof we needto assume { F j } i.i.d. Assume that both processes { F j } and { G j } satisfy the assumptions madein Theorem 2.2. With some abuse of notation we label the time constants, means, and evenrandom weights associated to the processes { F j } and { G j } with subscripts F and G . So forexample µ F = E ( R x dF ( x )). The symbolic subscripts F and G should not be confused with therandom distributions F j and G j assigned to the rows of the lattice. We write Ψ Ber ([ G ( x ) − F ( x )] + ) for the limit of a Bernoulli model with weight distributions P ( X ( i, j ) = 1) = ( G j ( x ) − F j ( x )) + =1 − P ( X ( i, j ) = 0) where x is a fixed parameter. An analogous convention will be used for otherBernoulli models along the way. Lemma 5.1.
Assume { F j } and { G j } satisfy (2.2) , (2.3) , (2.5) and (2.6) . Then for α > , | Ψ F ( α, − Ψ G ( α, − ( µ F − µ G ) |≤ √ α Z + ∞−∞ (cid:16) E | G ( x ) − F ( x ) | (cid:17) / dx + α Z + ∞−∞ ess sup P | F ( x ) − G ( x ) | dx. (5.1) Proof.
The right-hand side of (5.1) is finite under the assumptions on { F j } and { G j } . Couplethe F j and G j distributed weights in a standard way. Let { u ( z ) : z = ( i, j ) ∈ Z } be i.i.d.Uniform(0 ,
1) random variables. Set X F ( z ) = F − j ( u ( z )), where F − j ( u ) = sup { x : F j ( x ) < u } ,and similarly X G ( z ) = G − j ( u ( z )). Write E for expectation over the entire probability space of AST-PASSAGE SHAPE 13 distributions and weights.Ψ F ( α, − Ψ G ( α, n →∞ n E max π ∈ Π( ⌊ αn ⌋ ,n ) X z ∈ π X F ( z ) − lim n →∞ n E max π ∈ Π( ⌊ αn ⌋ ,n ) X z ∈ π X G ( z ) ≤ lim n →∞ n E max π ∈ Π( ⌊ αn ⌋ ,n ) X z ∈ π (cid:0) X F ( z ) − X G ( z ) (cid:1) = lim n →∞ n E max π ∈ Π( ⌊ αn ⌋ ,n ) X z ∈ π Z + ∞−∞ n I (cid:0) X G ( z ) ≤ x < X F ( z ) (cid:1) − I (cid:0) X F ( z ) ≤ x < X G ( z ) (cid:1)o dx ≤ lim n →∞ n E Z + ∞−∞ max π ∈ Π( ⌊ αn ⌋ ,n ) X z ∈ π n I (cid:0) X G ( z ) ≤ x < X F ( z ) (cid:1) − I (cid:0) X F ( z ) ≤ x < X G ( z ) (cid:1)o dx. We check that Fubini allows us to interchange the integral and the expectation. Since F and G are interchangeable it is enough to consider the first indicator function from above. Let a be aninteger ≥ α . Z + ∞−∞ n E max π ∈ Π( ⌊ αn ⌋ ,n ) X z ∈ π I (cid:0) X G ( z ) ≤ x < X F ( z ) (cid:1) dx ≤ Z + ∞−∞ sup n n E max π ∈ Π( an,n ) X z ∈ π I (cid:0) X G ( z ) ≤ x < X F ( z ) (cid:1) dx = Z + ∞−∞ Ψ Ber ([ G ( x ) − F ( x )] + ) ( a, dx ≤ Z + ∞−∞ (cid:16) E | G ( x ) − F ( x ) | + 4 √ a (cid:0) E | G ( x ) − F ( x ) | (cid:1) / + a ess sup P | G ( x ) − F ( x ) | (cid:17) dx< ∞ by estimate (3.7) and the finiteness of the right-hand side of (5.1). Continue from the limitabove by applying Fubini. Then take the limit inside the dx -integral by dominated convergence,justified by the n -uniformity in the bound above. Finally apply again the Bernoulli estimate(3.7). Ψ F ( α, − Ψ G ( α, ≤ lim n →∞ Z + ∞−∞ n E max π ∈ Π( ⌊ αn ⌋ ,n ) X z ∈ π n I (cid:0) X G ( z ) ≤ x < X F ( z ) (cid:1) − I (cid:0) X F ( z ) ≤ x < X G ( z ) (cid:1)o dx ≤ Z + ∞−∞ lim n →∞ n n E max π ∈ Π( ⌊ αn ⌋ ,n ) X z ∈ π I (cid:0) X G ( z ) ≤ x < X F ( z ) (cid:1) + E max π ∈ Π( ⌊ αn ⌋ ,n ) X z ∈ π (cid:0) − I (cid:0) X F ( z ) ≤ x < X G ( z ) (cid:1)(cid:1) − X z ∈ π o dx = Z + ∞−∞ (cid:8) Ψ Ber ([ G ( x ) − F ( x )] + ) ( α,
1) + Ψ
Ber (1 − [ F ( x ) − G ( x )] + ) ( α, − (1 + α ) (cid:9) dx ≤ Z + ∞−∞ (cid:26) E (cid:0) G ( x ) − F ( x ) (cid:1) + + 1 − E (cid:0) F ( x ) − G ( x ) (cid:1) + + 4 √ α (cid:16) q E (cid:0) G ( x ) − F ( x ) (cid:1) + + q E (cid:0) F ( x ) − G ( x ) (cid:1) + (cid:17) + α (cid:16) ess sup P [ G ( x ) − F ( x )] + + 1 − ess inf P [ F ( x ) − G ( x )] + (cid:17) − (1 + α ) (cid:27) dx ≤ ( µ F − µ G ) + 8 √ α Z + ∞−∞ p E | F ( x ) − G ( x ) | dx + α Z + ∞−∞ ess sup P | G ( x ) − F ( x ) | dx. Interchanging F and G gives the bound from the other direction and concludes the proof. (cid:3) For a while we make two convenient assumptions: that the weights are uniformly bounded,so for a constant
M < ∞ ,(5.2) P { F ( − M ) = 0 and F ( M ) = 1 } = 1 , and that variances are uniformly bounded away from zero, so for a constant 0 < c < ∞ ,(5.3) P { σ ( F ) ≥ c } = 1 . Note that then(5.4) c ≤ σ ( F ) ≤ M P -a.sand the conditions assumed for Theorem 2.2 are trivially satisfied by the uniform boundedness.Henceforth r = r ( α ) denotes a positive integer-valued function such that r ( α ) ր ∞ as α ց × r blocks B r ( x, y ) = { ( x, ry + k ) : k = 0 , , ..., r − } for ( x, y ) ∈ Z . Acoarse-grained last-passage model is defined by adding up the weights in each block: X r ( z ) = X v ∈ B r ( z ) X ( v ) . The distribution of the new weight X r ( i, j ) on row j ∈ Z + of the rescaled lattice is the convolution F r, j = F rj ∗ F rj +1 ∗ · · · ∗ F rj + r − . We repeat Lemma 4.4 from [14] with a sketch of the argument.
Lemma 5.2.
Let Ψ F ( x, y ) and Ψ F r ( x, y ) be the last passage time functions obtained by using F j and F r,j as the distributions on the j th row, respectively. If r → ∞ and r √ α → as α ↓ ,then lim α ↓ √ α (cid:12)(cid:12) Ψ F ( α, − r Ψ F r ( αr, (cid:12)(cid:12) = 0 . Proof.
Given a path π ∈ Π( m, nr − π ∈ Π( m, n −
1) in the rescaled lattice such that (cid:12)(cid:12) ( ∪ z ∈ ˜ π B r ( z )) △ π (cid:12)(cid:12) ≤ mr. Then by (5.2) (cid:12)(cid:12) max π ∈ Π( m,nr ) X z ∈ π X ( z ) − max ˜ π ∈ Π( m,n ) X z ∈ ˜ π X r ( z ) (cid:12)(cid:12) ≤ mrM. Take m = ⌊ αnr ⌋ , divide through by nr , and the conclusion follows. (cid:3) Let µ r,y and V r,y be the mean and variance of F r,y : µ r,y = r − X i =0 µ ry + i , and V r,y = r − X i =0 σ ry + i . Let Φ r,y be the distribution function of the normal N ( µ r,y , V r,y ) distribution, and e Φ r,y thedistribution function of N ( rµ F , V r,y ). The difference between Φ r,y and e Φ r,y is that the latter hasa non-random mean. We shall also find it convenient to use { X j } as a sequence of independentvariables with (random) distributions X j ∼ F j . For the next lemma we need to assume { F j } ani.i.d. sequence under P .As in [14], a key step in the proof is the replacement of the rescaled weights with Gaussianweights, which is undertaken in the next lemma. AST-PASSAGE SHAPE 15
Lemma 5.3.
Assume { F j } i.i.d. under P . If r → ∞ and r √ α → as α ↓ , then (5.5) lim α ↓ r √ α | Ψ F r ( αr, − Ψ Φ r ( αr, | = 0 . Proof.
According to Theorem 5.17 of [15], independent mean 0 random variables X , X , X , . . . satisfy the estimate (cid:12)(cid:12)(cid:12) P n B − / r r X i =1 X i ≤ x o − Φ( x ) (cid:12)(cid:12)(cid:12) ≤ A P ri =1 E | X i | B / r (1 + | x | ) − , x ∈ R , where B r = P ri =1 Var( X i ), Φ is the standard normal distribution function, and A is a constantthat is independent of the distribution functions of X , X , . . . , X r . Then,(5.6) | F r,y ( x ) − Φ r,y ( x ) | ≤ A P r − i =0 E | X ry + i − µ ry + i | ( P r − i =0 σ ry + i ) / (cid:0) V − / r,y | x − µ r,y | (cid:1) − ≤ C √ r (cid:0) M − r − / | x − µ r,y | (cid:1) − where the second inequality used the assumptions P ( | X i | ≤ M ) = 1 and σ i ≥ c > { F r,y } y ∈ Z + and { Φ r,y } y ∈ Z + and with α replaced by αr .For the first term on the right in (5.1), note this Schwarz trick: for a probability density f on R and a function H ≥ Z √ H dx = Z f / p f − H dx ≤ (cid:18) Z f − H dx (cid:19) / . For the calculation below take δ > f ( x ) = c (1 + | x − rµ F | δ ) − for the right constant c = c ( δ ). Factors that depend on M and δ are subsumed in a constant C . Then √ αr Z + ∞−∞ (cid:16) E | F r, ( x ) − Φ r, ( x ) | (cid:17) / dx ≤ Cα / r / Z + ∞−∞ (cid:26) E h(cid:0) M − r − / | x − µ r, | (cid:1) − i(cid:27) / dx ≤ Cα / r / (cid:26) E Z + ∞−∞ (cid:0) | x − rµ F | δ (cid:1)(cid:18) | x − µ r, | M √ r (cid:19) − dx (cid:27) / by a change of variables x = µ r, + yM √ r = Cα / r / (cid:26) E Z + ∞−∞ | µ r, − rµ F + yM √ r | δ (1 + | y | ) dy (cid:27) / ≤ Cα / r / n E | µ r, − rµ F | δ + r (1+ δ ) / o / ≤ Cα / r (3+ δ ) / . In the last step we used E | µ r, − rµ F | δ ≤ Cr (1+ δ ) / which follows because µ r, − rµ F is a sumof r bounded mean zero i.i.d. random variables. For the second term on the right in (5.1), αr Z + ∞−∞ ess sup P | F r, ( x ) − Φ r, ( x ) | dx ≤ Cαr / Z + ∞−∞ ess sup P (cid:18) | x − µ r,y | M √ r (cid:19) − dx ≤ Cαr / (cid:26)Z − rM −∞ (cid:18) − rM − xM √ r (cid:19) − dx + Z rM − rM dx + Z + ∞ rM (cid:18) x − rMM √ r (cid:19) − dx (cid:27) ≤ Cαr / . To summarize, with these estimates and (5.1) we have1 r √ α | Ψ F r ( αr, − Ψ Φ r ( αr, | ≤ Cr √ α ( α / r (3+ δ ) / + αr / ) . If δ is fixed small enough, assumptions r → ∞ and r √ α → α → (cid:3) The next lemma makes a further approximation that puts us in the situation where all siteshave normal variables with the same mean.
Lemma 5.4.
Let Ψ Φ r and Ψ e Φ r be defined as before, and again r √ α → as α → . Then lim α ↓ r √ α | Ψ Φ r ( αr, − Ψ e Φ r ( αr, | = 0 . Proof.
For z = ( i, j ) ∈ Z , let X ( r ) ( z ) have distribution Φ r,j so that e X ( r ) ( z ) = X ( r ) ( z ) − µ r,j + rµ F has distribution e Φ r,j . Now estimate:Ψ e Φ r ( αr,
1) = lim n →∞ n max π ∈ Π( ⌊ αnr ⌋ ,n ) X z ∈ π e X ( r ) ( z ) ≤ lim n →∞ n max π ∈ Π( ⌊ αnr ⌋ ,n ) X z ∈ π X ( r ) ( z ) + lim n →∞ n max π ∈ Π( ⌊ αnr ⌋ ,n ) X z ∈ π (cid:0) − µ r,j + rµ F (cid:1) ≤ Ψ Φ r ( αr,
1) + lim n →∞ n n X j =1 (cid:0) − µ r,j + rµ F (cid:1) + lim n →∞ n M r · ⌊ αnr ⌋ = Ψ Φ r ( αr,
1) + 2
M αr . The opposite bound Ψ e Φ r ( αr, ≥ Ψ Φ r ( αr, − M αr comes similarly, and the lemma follows. (cid:3) Let us separate the mean by letting Φ r,y denote the N (0 , P r − i =0 σ ry + i ) distribution function.Since the last-passage functions of the normal distributions satisfy Ψ e Φ r ( αr,
1) = rµ F (1 + αr ) +Ψ Φ ( r ) ( αr, Lemma 5.5.
Assume { F j } i.i.d. under P , and assume r = r ( α ) satisfies r → ∞ and r √ α → as α ↓ . Under assumptions (5.2) and (5.3)(5.7) lim α ↓ √ α | Ψ F ( α, − µ F − r Ψ Φ ( r ) ( αr, | = 0 . In order to deduce a limit from (5.7) we utilize the explicitly computable case of exponentialdistributions from [20]. We need to match up the random variances of the exponentials withthe variances σ j of the sequence { F j } . Thus, given the i.i.d. sequence of quenched variances σ j = σ ( F j ) that we have worked with up to now under condition (5.4), let ξ j = 1 /σ j and G j ( x ) = 1 − e − ξ j x the rate ξ j exponential distribution. Then { ξ j } j ∈ Z + is an i.i.d. sequence of AST-PASSAGE SHAPE 17 bounded random variables 0 < c ≤ ξ j ≤ b with distribution m . We can assume c is the exactlower bound: m [ c, c + ε ) > ε > G j has mean and variance µ ( G j ) = ξ − j and σ ( G j ) = ξ − j = σ j .Assumptions (2.2) and (2.3) are easily checked, and so the last-passage function Ψ G is well-defined. We would like to apply Lemma 5.5 to this exponential model, but obviously assumption(5.2) is not satisfied. To get around this difficulty we do the following approximation whichleaves the quenched means and variances intact. We learned this trick from [14].Let Y j denote a G j -distributed random variable. For a fixed τ >
0, let m j = E ( Y j | Y j > τ ) and w j = E ( Y j | Y j > τ ) . The quantities s j = ( m j − τ ) ( m j − τ ) + w j − m j and u j = w j − τ m j − τ − τ satisfy the equations (1 − s j ) τ + s j u j = m j and (1 − s j ) τ + s j u j = w j . Then 0 ≤ s j ≤ u j ≥ τ and w j ≥ τ . Define distribution functions(5.8) e G j ( x ) = G j ( x ) 0 ≤ x < τ − s j [1 − G j ( τ )] τ ≤ x < u j x ≥ u j . e Y j ∼ e G j satisfies EY j = E e Y j and EY j = E e Y j . Moreover, for any fixed τ > u j = E ( Y j | Y j > τ ) − τ E ( Y j | Y j > τ ) − τ − τ = 2 p j + τ ≤ c + τ, so the distributions { e G j } are all supported on the nonrandom bounded interval [0 , /c + τ ].Consequently Lemma 5.5 applies to e G . We can draw the same conclusion for G once we havethe next estimate: Lemma 5.6.
Given ε > , we can select τ large enough and define e G j as in (5.8) so that lim α ↓ √ α | Ψ G ( α, − Ψ e G ( α, | < ε. Proof.
This comes from an application of Lemma 5.1. G j = e G j on ( −∞ , τ ) and 1 − e G j ≤ − G j on all of R . The integrals on the right-hand side of (5.1) are finite and can be made arbitrarilysmall by choosing τ large. (cid:3) Currently we have shown that(5.9) lim α ↓ √ α | Ψ G ( α, − E σ − r Ψ Φ r ( αr, | = 0 . It remains to perform an explicit calculation on Ψ G ( α, µ G = E ξ − and σ G = E ξ − . Lemma 5.7.
For random exponential distributions with rates bounded away from zero, Ψ G ( α,
1) = µ G − σ G √ α + O ( α ) . Proof.
Recall the definition of the limit shape Ψ G ( α,
1) from (2.10). From (2.9) one can readthat tg (1 /t ) is nondecreasing in t . Thus by (2.10) Ψ G ( α,
1) = t = t ( α ) such that tg (1 /t ) = α .Next we argue that when α is close enough to 0, g (1 /t ) = − u /t + a ( u ) for some 0 < u < u ∗ with a ′ ( u ) = 1 /t . Since a (0) = 0 and a ( u ∗ − ) = c , strict concavity gives for 0 < u < u ∗ nZ [ c, ∞ ) ξ ( ξ − c ) m ( dξ ) o − = a ′ ( u ∗ − ) < a ′ ( u ) = nZ [ c, ∞ ) ξ ( ξ − a ( u )) m ( dξ ) o − < a ′ (0+) = nZ [ c, ∞ ) ξ − m ( dξ ) o − = 1 µ G . On the other hand, 0 < Ψ G ( α, − µ G ≤ C √ α + Cα where the second inequality comes fromcomparing { G j } in (5.1) with identically zero weights. Thus when α is small enough, 1 /t is inthe range of a ′ . Consequently there exists u ∈ (0 , u ∗ ) such that a ′ ( u ) = 1 /t , or equivalently,(5.10) Z [ c, ∞ ) ξ ( ξ − a ( u )) m ( dξ ) = t. From the choice of t , α = tg (1 /t ) = t (cid:0) − u /t + a ( u ) (cid:1) = − u + ta ( u ) and so(5.11) Ψ G ( α,
1) = t = αa ( u ) + u a ( u ) = αa ( u ) + Z [ c, ∞ ) ξ − a ( u ) dm ( ξ ) . Combining (5.10) and (5.11) gives(5.12) α = a ( u ) Z [ c, ∞ ) ξ − a ( u )) m ( dξ ) . From this a ( u ) σ G = a ( u ) Z [ c, ∞ ) ξ m ( dξ ) ≤ α. Hence we have 0 ≤ a ( u ) ≤ √ α/σ G . When α and hence a ( u ) is small, (5.12) and the last bound on a ( u ) yield0 ≤ αa ( u ) − σ G = Z [ c, ∞ ) (cid:2) ξ − a ( u )) − ξ (cid:3) m ( dξ )= Z [ c, ∞ ) ξa ( u ) − a ( u ) ξ ( ξ − a ( u )) m ( dξ ) ≤ Z [ c, ∞ ) a ( u ) ξ ( c − a ( u )) m ( dξ ) = O ( √ α ) . (5.13)Consequently √ αa ( u ) − σ G = αa ( u ) − σ G √ αa ( u ) + σ G = O ( √ α ) . AST-PASSAGE SHAPE 19
Now we put all the above together to prove the lemma.Ψ G ( α, − µ G − σ G √ α = αa ( u ) + Z [ c, ∞ ) ξ − a ( u ) m ( dξ ) − µ G − σ G √ α = αa ( u ) + Z [ c, ∞ ) h ξ + 1 ξ a ( u ) + O (cid:0) a ( u ) (cid:1)i m ( dξ ) − µ G − σ G √ α = √ α (cid:16) √ αa ( u ) − σ G (cid:17) + σ G a ( u ) (cid:16) σ G − √ αa ( u ) (cid:17) + α · O (cid:16) a ( u ) α (cid:17) = O ( α ) as α ↓ (cid:3) Combining Lemma 5.7 and (5.9) giveslim α ↓ √ α (cid:12)(cid:12) r Ψ Φ r ( αr, − σ G √ α (cid:12)(cid:12) = 0 . Substitute this back into (5.7) and recall that σ F = σ G . The conclusion we get is(5.14) lim α ↓ √ α | Ψ F ( α, − µ F − σ F √ α | = 0 . We have proved Theorem 2.2 under assumptions (5.2) and (5.3). We now lift (5.3). For ε >
0, let { W ( z ) } be i.i.d weights with distribution H defined by P ( W ( z ) = ± ε ) = 1 /
2. Let e F j = F j ∗ H be the distribution of the weight e X ( i, j ) = X ( i, j ) + W ( i, j ). Let Ψ H and Ψ e F be thetime constants of the last-passage models with weights { W ( z ) } and { e X ( z ) } , respectively. TheBernoulli bound (3.7) gives the estimate Ψ H ( x, y ) ≤ ε √ xy . The corresponding last-passagetimes satisfy T e F ( z ) − T H ( z ) ≤ T F ( z ) ≤ T e F ( z ) + ˆ T H ( z )where ˆ T H ( z ) uses the weights − W ( z ). In the limit(5.15) Ψ e F ( α, − ε √ α ≤ Ψ F ( α, ≤ Ψ e F ( α,
1) + 4 ε √ α. Since σ ( e F j ) = σ ( F j ) + ε while µ e F = µ F , and ε > { F j } satisfy the conditions required for Theorem 2.2, but there is no commonbounded support. For a fixed M > F j,M ( x ) = x ≥ MF j ( x ) − M ≤ x < M x < − M. Let µ M , σ M and Ψ F M ( x, y ) be quantities associated to { F j,M } . From (5.1) and the conditions assumed in Theorem 2.2,1 √ α | Ψ F ( α, − Ψ F M ( α, − ( µ − µ M ) |≤ Z + ∞−∞ (cid:16) E | F ( x ) − F ,M ( x ) | (cid:17) / dx + √ α Z + ∞−∞ ess sup P | F ( x ) − F ,M ( x ) | dx = 8 hZ − M −∞ (cid:16) E | F ( x ) | (cid:17) / dx + Z ∞ M (cid:16) E | − F ( x ) | (cid:17) / dx i + √ α hZ − M −∞ ess sup P | F ( x ) | dx + Z + ∞ M ess sup P | − F ( x ) | dx i ≤ ε. The last inequality comes from choosing M large enough, and is valid for all α ≤
1. Since E ( EX (0 , < ∞ , dominated convergence gives σ M → σ and so we can pick M so that | σ − σ M | < ε . Now1 √ α | Ψ F ( α, − µ − σ √ α | ≤ √ α | Ψ F M ( α, − µ M − σ M √ α | + 2 ε. Since ε is arbitrary and limit (5.14) holds for { F j,M } , we get the conclusion for the sequence { F j } . This concludes the proof of Theorem 2.2.6. Proof of Theorem 2.3
Proof of Theorem 2.3.
The lower bound in (2.8) can be proved by applying Martin’s result (1.2)to the homogeneous problem where a maximal path is constructed by using only those rows j where F j = H i ∗ , the distribution with the maximal mean µ ∗ = µ ( H i ∗ ). This is fairly straight-forward and we leave the details to the reader.To prove the upper bound in (2.8), we start by increasing all the weights X ( z ) by movingtheir means to µ ∗ . Then we subtract the common mean µ ∗ from the weights, so that for theproof we can assume that all distributions H , . . . , H L have mean zero. Create the following coupling. Independently of the process { F j } , let { X ℓ ( z ) : 1 ≤ ℓ ≤ L, z ∈ Z } be a collection of independent weights such that X ℓ ( z ) has distribution H ℓ . Then definethe weights used for computing Ψ(1 , α ) by X ( z ) = L X ℓ =1 X ℓ ( z ) I { F j = H ℓ } for z = ( i, j ) ∈ Z .Begin with this elementary bound:(6.1) Ψ(1 , α ) = lim n →∞ n E h max π ∈ Π( n, ⌊ αn ⌋ ) X z ∈ π X ( z ) i ≤ L X ℓ =1 lim n →∞ n E h max π ∈ Π( n, ⌊ αn ⌋ ) X z ∈ π X ℓ ( z ) I { F j = H ℓ } i . The next lemma contains a convexity argument that will remove the indicators from the lastpassage values above.
Lemma 6.1.
Let D be a sub- σ -field on a probability space (Ω , F , P ) , D an event in D , and ξ and η two integrable random variables. Assume that Eη = 0 , η is independent of D , and ξ and η are independent conditionally on D . Then E [ ξ ∨ ( ηI D ) ] ≤ E [ ξ ∨ η ] . AST-PASSAGE SHAPE 21
Proof.
By Jensen’s inequality, for any fixed x ∈ R , x ∨ E ( η | D ) ≤ E ( x ∨ η | D ) . Since η is independent of D and mean zero, x ∨ ≤ E ( x ∨ η | D ) . Integrate this against the conditional distribution P ( ξ ∈ dx | D ) of ξ , given D , and use theconditional independence of ξ and η : E ( ξ ∨ | D ) ≤ E ( ξ ∨ η | D ) . Next integrate this over the event D c : E (cid:2) I D c · ξ ∨ ( ηI D ) (cid:3) = E (cid:2) I D c · ξ ∨ (cid:3) ≤ E (cid:2) I D c · ξ ∨ η (cid:3) . The corresponding integral over the event D needs no argument. (cid:3) Fix a lattice point z = ( i , j ) for the moment. We split the maximum in (6.1) according towhether the path π goes through z or not, and in case it goes we also separate the weight at z :max π ∈ Π( n, ⌊ nα ⌋ ) X z ∈ π X ℓ ( z ) I { F j = H ℓ } = A ∨ (cid:0) B + X ℓ ( z ) I { F j = H ℓ } (cid:1) = B + (cid:0) A − B (cid:1) ∨ (cid:0) X ℓ ( z ) I { F j = H ℓ } (cid:1) where A = max π z X z ∈ π X ℓ ( z ) I { F j = H ℓ } and B = max π ∋ z X z ∈ π \{ z } X ℓ ( z ) I { F j = H ℓ } . Now apply Lemma 6.1 with ξ = A − B , η = X ℓ ( z ), and D = { F j = H ℓ } . Given F j , A − B does not look at X ℓ ( z ), so the independence assumed in Lemma 6.1 is satisfied. The outcomefrom that lemma is the inequality E h max π ∈ Π( n, ⌊ αn ⌋ ) X z ∈ π X ℓ ( z ) I { F j = H ℓ } i ≤ E (cid:2) A ∨ ( B + X ℓ ( z )) (cid:3) . This is tantamount to replacing the weight X ℓ ( z ) I { F j = H ℓ } at z with X ℓ ( z ).We can repeat this at all lattice points z in (6.1). In the end we have an upper bound interms of homogeneous last-passage values, to which we can apply Martin’s result (1.2):Ψ(1 , α ) ≤ L X ℓ =1 lim n →∞ n E h max π ∈ Π( n, ⌊ αn ⌋ ) X z ∈ π X ℓ ( z ) i = L X ℓ =1 Ψ H ℓ (1 , α )= 2 √ α L X ℓ =1 σ ( H ℓ ) + o ( √ α ) . This completes the proof of Theorem 2.3. (cid:3) Proofs for the exponential model
Proof of Theorem 2.4.
Equation (2.10) gives(7.1) Ψ G (1 , α ) = inf { t ≥ tg ( α/t ) ≥ } = t ( α ) = t. That the infimum is achieved can be seen from (2.9).Under Case 1 the critical value u ∗ = R [ c, ∞ ) c ( ξ − c ) − m ( dξ ) < ∞ , and also a ′ ( u ∗ − ) = (cid:26)Z [ c, ∞ ) ξ ( ξ − c ) m ( dξ ) (cid:27) − > . By the concavity of a and (2.9), for 0 ≤ y ≤ a ′ ( u ∗ − ) we have g ( y ) = − yu ∗ + c . Consequentlyfor small enough α tg ( α/t ) = − αc Z [ c, ∞ ) ξ − c m ( dξ ) + ct and equation (2.12) follows.In Case 2 a ′ (0+) > a ′ ( u ∗ − ) = 0 and hence for small enough α > u ∈ (0 , u ∗ ) such that a ′ ( u ) = α/t . Set a = a ( u ) ∈ (0 , c ). As α ց
0, both u ր u ∗ and a ր c . We have the equations a ′ ( u ) − = Z [ c, ∞ ) ξ ( ξ − a ) m ( dξ ) = tα , tg ( α/t ) = − αu + ta , (7.2) Ψ G (1 , α ) = t = 1 a + αu a = 1 a + α Z [ c, ∞ ) ξ − a m ( dξ )and(7.3) 1 a = α Z [ c, ∞ ) ξ − a ) m ( dξ ) . Assuming (2.11), start with ν ∈ ( − , ∪ (0 , ε > < κ < κ such that(7.4) κ ( ξ − c ) ν +1 ≤ m [ c, ξ ) ≤ κ ( ξ − c ) ν +1 for ξ ∈ [ c, c + ε ]and as ε ց κ , κ → κ . First we estimate c − a . Fix ε > α = a Z [ c, ∞ ) ξ − a ) m ( dξ ) = 2 a Z ∞ c m [ c, ξ )( ξ − a ) dξ = 2 a Z c + εc m [ c, ξ )( ξ − a ) dξ + C ( ε )for a quantity C ( ε ) = O ( ε − ). The first term above can be bounded above and below by (7.4),and we develop both bounds together for κ i , i = 1 ,
2, as(7.5) 2 κ i a Z c + εc ( ξ − c ) ν +1 ( ξ − a ) dξ + C ( ε )= 2 κ i a Z c + εc (cid:2) ( ξ − a ) − ( c − a ) (cid:3) ν +1 ( ξ − a ) dξ + C ( ε )= 2 κ i a ∞ X k =0 (cid:18) ν + 1 k (cid:19) ( − k ( c − a ) k Z c + εc ( ξ − a ) ν − k − dξ + C ( ε )= 2 κ i a ∞ X k =0 (cid:18) ν + 1 k (cid:19) ( − k ( c − a ) k ( c − a ) ν − k − − ( c + ε − a ) ν − k − k − ν + 1 + C ( ε )= 2 κ i a A ν ( c − a ) ν − − κ i a ∞ X k =0 (cid:18) ν + 1 k (cid:19) ( − k k − ν + 1 ( c − a ) k ( c + ε − a ) ν − k − + C ( ε )= 2 κ i a A ν ( c − a ) ν − + C ( ε ) .C ( ε ) changed of course in the last equality. In the next to last equality above we defined A ν = ∞ X k =0 (cid:18) ν + 1 k (cid:19) ( − k k − ν + 1 . AST-PASSAGE SHAPE 23
Rewrite the above development in the form( c − a ) − ν = 2 κc A ν α + α [2 A ν ( κ i a − κc ) + C ( ε )( c − a ) − ν ] . Now choose ε = ε ( α ) ց α ց C ( ε )( c − a ) − ν → α ց κ i a → κc and we can write(7.6) c − a = B α − ν + o ( α − ν )with a new constant B = (2 κc A ν ) − ν .Now consider the case ν ∈ (0 ,
1) which also guarantees R [ c, ∞ ) ( ξ − c ) − m ( dξ ) < ∞ . From (7.2)and (7.6) as α ց G (1 , α ) = 1 a + α Z [ c, ∞ ) ξ − a m ( dξ )= 1 c + α Z [ c, ∞ ) ξ − c m ( dξ ) + O ( α − ν ) + α (cid:18) Z [ c, ∞ ) ξ − a m ( dξ ) − Z [ c, ∞ ) ξ − c m ( dξ ) (cid:19) = 1 c + α Z [ c, ∞ ) ξ − c m ( dξ ) + o ( α ) . Next the case ν ∈ ( − , G (1 , α ) = 1 a + α Z [ c, ∞ ) ξ − a m ( dξ )= 1 c + c − a c + ( c − a ) c a + α Z c + εc m [ c, ξ )( ξ − a ) dξ + αC ( ε ) . Again, using (7.4) and proceeding as in (7.5), we develop an upper and a lower bound for thequantity above with distinct constants κ i , i = 1 ,
2. After bounding m [ c, ξ ) above and below with κ i ( ξ − c ) ν +1 in the integral, write ( ξ − c ) ν +1 = (( ξ − a ) − ( c − a )) ν +1 and expand in powerseries. 1 c + B c − α − ν + o ( α − ν ) + ακ i Z c + εc ( ξ − c ) ν +1 ( ξ − a ) dξ + αC ( ε )= 1 c + B c − α − ν + o ( α − ν ) + ακ i ( c − a ) ν ∞ X k =0 (cid:18) ν + 1 k (cid:19) ( − k k − ν + ακ i ( c − a + ε ) ν ∞ X k =0 (cid:18) ν + 1 k (cid:19) ( − k ν − k (cid:18) c − a c − a + ε (cid:19) k + αC ( ε )= 1 c + Bα − ν + o ( α − ν ) + A ν, α ( κ i − κ )( c − a ) ν + αC ( ε ) . In the last equality the next to last term with the P ∞ k =0 sum was subsumed in the αC ( ε ) term.Then we introduced new constants(7.7) A ν, = ∞ X k =0 (cid:18) ν + 1 k (cid:19) ( − k k − ν and B = B c − + κB ν A ν, . As before, by letting ε = ε ( α ) ց α ց G (1 , α ) = c − + Bα − ν + o ( α − ν ) from the above bounds.It remains to treat the cases ν = − , , (cid:3) References [1] E. D. Andjel, P. A. Ferrari, H. Guiol, and C. Landim. Convergence to the maximal invariant measure for azero-range process with random rates.
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Hao Lin, Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, USA
E-mail address : [email protected] Timo Sepp¨al¨ainen, Department of Mathematics, University of Wisconsin-Madison, Madison, WI53706, USA
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