Limit shapes for inhomogeneous corner growth models with exponential and geometric weights

We generalize the exactly solvable corner growth models by choosing the rate of the exponential distribution $a_i+b_j$ and the parameter of the geometric distribution $a_i b_j$ at site $(i, j)$, where $(a_i)_{i \ge 1}$ and $(b_j)_{j \ge 1}$ are jointly ergodic random sequences. We identify the shape function in terms of a simple variational problem, which can be solved explicitly in some special cases.


Introduction
A (planar) corner growth model describes the random evolution over time of a growing cluster on a quadrant of the plane, [16,Chapter 1]. We represent the quadrant with N 2 , where N " t1, 2, . . . u, and the cluster with a family of subsets St Ă N 2 indexed by time t ě 0. Let W " tW pi, jq : i, j P Nu be a collection of nonnegative real-valued random variables with joint probability distribution P . The site pi, jq P N 2 joins the cluster after a waiting time of W pi, jq since the first time both of the following conditions hold: • If i ą 1 the neighbor pi´1, jq is in the cluster.
• If j ą 1 the neighbor pi, j´1q is in the cluster.

In other words,
St " tpi, jq P N 2 : Gpi, jq ď tu for t ě 0, where the function G is defined through the recursion Gpi, jq " Gpi´1, jq _ Gpi, j´1q`W pi, jq for i, j P N (1.1) for all 1 ď k ă n. In this case, π is said to be from pi1, j1q to pin, jnq. As first observed in [12], the solution G of recursion (1.1) subject to boundary conditions (1.2) is given by for M, N P N, where the maximum is taken over all directed paths π from p1, 1q to pM, N q.
The quantity on the right-hand side of (1.3) is called the last-passage time from p1, 1q to pM, N q in the context of directed last-passage percolation, [11]. In order to understand the behavior of the cluster tSt : t ě 0u, it is of interest to study the random variables tGpi, jq : i, j P Nu. When it is possible to obtain explicit information about various statistics of these random variables, the P -model is regarded as exactly solvable.
In this paper, we consider certain classes of exactly solvable models and describe the leading order asymptotics of Gpr N x s, r N y sq as N Ñ 8, where x, y ą 0 are fixed.
We restrict attention to the P -models for which the limit Gpr N x s, r N y sq N (1. 4) exists and is deterministic for all x, y ą 0, P -almost surely. Following [11], we call the function gP : p0, 8q 2 Ñ r0, 8s the shape function of the P -model. More specifically, we look at the following two kinds of models: • Exponential model. Let a " panq nPN and b " pbnq nPN be sequences in D " p0, 8q. Suppose that the waiting times are independent and the marginal distribution at pi, jq is exponential with parameter ai`bj for each i, j P N. That is, P pW pi, jq ě xq " expp´pai`bjqxq for x ě 0. (1.5) • Geometric model. Let a " panq nPN and b " pbnq nPN be sequences in D " p0, 1q. Suppose that the waiting times are independent and the marginal distribution at pi, jq is geometric with fail parameter aibj for each i, j P N. That is, P pW pi, jq " kq " p1´aibjqa k i b k j for k " 0, 1, . . . (1.6) We refer to a and b as the column and row parameters, respectively. To distinguish these models from an arbitrary corner growth model, we will write P a,b instead of P for the measure. Note our convention of employing the same notation for analogous objects associated with exponential and geometric models. This is for convenience as we will treat both models in the same fashion. When there is need to indicate which model is under consideration, we will sometimes decorate the notation with superscripts exp or geom appropriately.
The problem we take up is to identify the shape function for the exponential and geometric models. Our main result is a variational formula characterizing the shape function for typical choice of the parameters a and b. We state this more precisely. Suppose that a and b are random with a joint distribution µ that is stationary with respect to the shifts ppanq nPN , pbnq nPN q Þ Ñ ppa n`k q nPN , pb n`l q nPN q for k, l P Z`and ergodic separately with respect to the shifts with k, l ą 0. In particular, a and b can be i.i.d. and independent of each other. Let us write P and Q for the marginal distributions of a and b, respectively. Then for µ-almost every a and b, the function g P a,b exists and satisfies g P a,b px, yq " inf zPI txApzq`yBpzqu for x, y ą 0, (1.7) where I is an interval (or a point in certain cases), and A and B are explicit functions determined by P and Q, see section 2 for definitions. By examining (1.7), we observe that g P a,b exhibits linear and nonlinear behavior in different sectors of the first quadrant depending on the tail of P and Q. Finally, we derive explicit formulas for g P a,b for certain choices of P and Q. An example is the exponential model when P and Q are taken as uniform measures on the intervals rc{2, c{2`Ls and rc{2, c{2`M s, respectively, for some c, L, M ą 0. Then for x, y ą 0 where uptq " 2tpc`M q L´tM`apL´tM q 2`4 tpc`M qpc`Lq for t ą 0. We consider a few more examples in section 2.
The geometric model appears in [5] and [6], and the exponential model is the continuous analogue. The notable feature present in both is sitewise inhomogeneity, that the distribution of W pi, jq depends on pi, jq. Furthermore, it turns out that both of these models are exactly solvable in the sense that the distribution of GpM, N q can be computed for each M, N P N. For the geometric model, see [6] for details. The exponential model can be treated as the limit of the geometric model in the following way: Let α " pαiq iPN and β " pβjq jPN denote the column and row parameters, respectively, of the exponential model. Choose the column and row parameters a " paiq iPN and b " pbjq jPN of the geometric model such that ai " 1´αi{n for i P rM s and bj " 1´βj{n for j P rN s for some sufficiently large n P N. Then it can be checked that P geom a,b pGpM, N q ď nxq Ñ P exp α,β pGpM, N q ď xq as n Ñ 8 for any x P R.
Our approach to the problem at hand is not based on the formulas for the distribution of the last passage times. Instead, we adapt a technique from [16], which treats the geometric model with i.i.d waiting times, that is, when a and b are constant sequences. The essence of the argument is to introduce a suitable family of stationary corner growth models on Z 2 , compute their shape functions and relate these functions to gP . Many steps in this scheme carry over to the more general exponential and geometric models introduced above with appropriate modifications to handle inhomogeneity. The main technical contribution of this paper is to extract gP from its relations to the shape functions of the stationary models.
We now begin a brief survey of the related results from the literature. An early work in the subject is due to Rost, [13]: If P is i.i.d with exponential marginals, then gP px, yq " mpx`yq`2σ ? xy (1.8) where m and σ are the common mean and the standard deviation of the waiting times. Formula (1.8) is also valid if P is i.i.d. with geometric marginals; a proof can be found in [16]. At present, these are the only two examples of exactly solvable models with i.i.d waiting times, for which it is possible to write down the shape function explicitly. We consider them as homogeneous models as they are obtained from the exponential and the geometric model by taking the parameters a and b as constant sequences. For more general i.i.d. waiting times, one has the following result established in [10]. If P is i.i.d with marginals subject to the condition 8 ż 0 a P pW p1, 1q ě sqds ă 8 (1.9) but otherwise arbitrary, then (1.8) approximately holds close to the boundary. The precise statement is that gP p1, αq " gP pα, 1q " m`2σ ? α`op ? αq as α Ó 0. The exponential model with only b chosen as a constant sequence, appears in [17] within a proof. In [9], the asymptotics of the shape function near boundary is investigated for more general models with inhomogeneity. Now P is a product measure such that the marginal distributions at the sites pi, jq and pi 1 , j 1 q are identical if j " j 1 . Then asymptotics of gP pα, 1q is similar to (1.9) but gP p1, αq behaves qualitatively differently as α Ó 0. These conclusions apply to almost every P randomly selected by drawing the marginal distributions for each row j from a distribution-valued ergodic process, see [9] for details.
Variational formulas have recently been obtained in [4] for homogeneous P -models when the waiting times have finite p-moment for some p ą 2. In section 4, we comment on some similarities between (1.7) and these formulas, which suggest that a suitable generalization of [4] would also give (1.7).
Corner growth models with exponential or geometric waiting times that feature inhomogeneity in a different way than considered in this paper recently appeared in [2], where the interest is also in characterization of the shape function. The parameters a and b are now determined by a function λ : r0, 8q 2 Ñ r0, 8q instead of being random. In our notation, the setup for the exponential version is as follows. For N P N, consider the PN -model where PN is the product measure such that for each i, j P N, the marginal at pi, jq is exponential distribution with mean λpi{N, j{N q. These models can be constructed on a common probability space. A result established in [2] is that, under some conditions on λ, limit p1.4q equals almost surely to the unique monotone viscosity solution of where p¨q`denotes the positive part.
We organize the paper as follows: Precise account of our results appears in section 2. In section 3, we note the existence of the shape functions for the exponential and geometric models. We devote section 4 to a discussion of certain families of stationary corner growth models on Z 2 . The proof of the main results are in section 5.

Results
We discuss our results in this section and illustrate them on a few examples. The method of computing the shape function is the same for the exponential and geometric models. Hence, we treat these two models simultaneously for the most part. On occasions when there are minor differences in the way the analysis proceeds, we consider the exponential model only and leave out the analogous arguments for the geometric model.
Recall that D exp " p0, 8q and D geom " p0, 1q. Let τ k : D N Ñ D N denote the shift map pcnq nPN Þ Ñ pc n`k q nPN for k P Z`. We equip D with the Borel σ-algebra, and D N and D NˆDN with the product σ-algebras. Let µ be a probability measure on D NˆDN that satisfies the following conditions for any event B Ă D NˆDN .
For i " 1, 2, let us write µi for the projection of µ onto the ith coordinate. It follows from the above assumptions on µ that µ1 and µ2 are ergodic measures on D N with respect to τ k for any k P N. We will sample the column and the row parameters a " panq nPN and b " pbnq nPN from the joint distribution µ. An example for µ is the product measure µ " µ1ˆµ2 where µ1 and µ2 are i.i.d. product measures; stationarity is due to the i.i.d. assumption and ergodicity can be deduced from Kolmogorov's 0-1 law.
Let P and Q denote the marginals of µ1 and µ2, respectively. Define Here, for a Borel measure P on D, supppP q denotes the support of P , which is the complement of the union of open sets in D of measure 0 under P . Note that α ď β.
Define functions A and B on pα,`8q and p´8, βq, respectively, by for the exponential model and for the geometric model. These will appear in the formulas for the shape function. Note that A exp pzq and B exp pzq are expectations of exponential random variables with parameters z`a and´z`b averaged over various a and b, respectively. Similarly, A geom pzq and B geom pzq are the averaged expectations of geometric random variables with (fail) parameters a{z and bz, respectively. The condition z P pα, βq ensures that the parameters take admissible values; that is, z`a exp ,´z`b exp P p0, 8q and a geom {z, b geom z P p0, 1q for all z P pα, βq.
Lemma 2.1. A and B are infinitely differentiable and the derivatives are given by for any k P N.
Proof. We obtain the first equality in (2.5) by induction on k; the others can be derived similarly. Suppose that the derivative pA exp q pk´1q exists and satisfies (2.5), which simply means (2.3) for k " 1. Also, A exp pzq and B exp pzq are finite for each z P pα, βq since the integrand in (2.3) is bounded by 1{pz´αq. For 0 ă |h| ă pz´αq{2, pz`a`hq j´k pz`aq´j´1 Ppdaq pz`aq´k´1 Ppdaq as h Ñ 0, by the dominated convergence theorem. Here, we dominated the integrands with the inequality pz`h`aq j´k pz`aq´j´1 ď pz´αq´k´1.
We will not need the derivatives of A and B of order higher than 2. In fact, the proofs of Theorems 2.2 and 2.3 stated below only require that A and B are continuous, A is decreasing and B is increasing. (The property of being a decreasing/increasing function is used in the strict sense throughout the paper.) The first two derivatives of A and B are used only in the present section to discuss some consequences of Theorem 2.2.
Theorem 2.2. Suppose that α ă β. Then, for µ-almost every choice of a and b, the shape function for the P a,b -model exists and satisfies g P a,b px, yq " inf zPpα,βq txApzq`yBpzqu (2.6) for x, y ą 0.
Note that Theorem 2.2 does not give information about the shape function of the P a,bmodel for a particular choice of a and b. A more ideal result would answer the following question for a given P a,b -model: Does the shape function exists and, if so, how does one describe it in terms of a and b?
Because of the symmetric roles of the columns and the rows in the corner growth model, one would expect that a formula for the shape function should be symmetric in P and Q. This is not readily apparent from (2.6); however, we can rewrite (2.6) in a form that brings out the symmetry of P and Q. Let us illustrate this for the exponential model.
In particular, if P and Q are the same then g P a,b px, yq " g P a,b py, xq for x, y ą 0. We now consider the case of α " β; this is when α " β " 0 for the exponential model and α " β " 1 for the geometric model. It turns out that then the situation is particularly simple and the shape function has an explicit formula. One simply needs to replace the interval pα, βq in (2.6) with the point tα " βu. Theorem 2.3. Suppose that α " β. Then, for µ-almost every choice of a and b, the shape function for the P a,b -model exists and satisfies the following: (a) For the exponential model Qpdbq for x, y ą 0. (2.7) (b) For the geometric model Qpdbq for x, y ą 0. (2.8) The integrals in (2.7) and (2.8) maybe infinite. The proofs of Theorem 2.2 and 2.3 involve two parts: We address the existence of g P a,b in section 3 and establish formulas (2.6), (2.7) and (2.8) in section 5.
Unless noted otherwise, we will focus attention to the case α ă β.
We now turn to implications of Theorem 2.2. Some notation is needed. Let us define A, B and their first derivatives A 1 , B 1 at the endpoints α and β as appropriate one-sided limits, which exist by monotonicity of these functions in view of Lemma 2.1. In fact, the values at the endpoints can be computed by substituting α and β for z in (2.5), by virtue of the monotone convergence theorem, since the integrands in (2.5) are all monotone in z. Thus, and so on. Note that, by Schwarz inequality, finiteness of A 1 pαq and B 1 pβq implies finiteness of Apαq and Bpβq, respectively. Let S denote the sector of the first quadrant on which B 1 pαq{A 1 pαq ă x{y ă´B 1 pβq{A 1 pβq holds, where the left-and right-hand sides are interpreted as 0 and`8 when A 1 pαq "´8 and B 1 pβq "`8, respectively. In addition, let S1 and S2 denote the sectors defined by the inequalities x{y ď´B 1 pαq{A 1 pαq and x{y ě´B 1 pβq{A 1 pβq, respectively. Possibly, S1 " H (if Corollary 2.4. Suppose that α ă β. Fix the parameters a and b such that the shape function of the P a,b -model exists and (2.6) holds. Let us also abbreviate g P a,b as g.
(e) g is continuously differentiable.
In particular, the shape function is homogeneous (property (a)) and concave. These properties are not specific to the exponential and geometric models, see the remarks preceding the proof of Proposition 3.1 below. Corollary 2.4 also shows that the shape function transitions smoothly between linear and strictly concave behavior if S1 or S2 is nonempty, as illustrated in Figure 2.1. It is expected for any homogeneous P -model that gP is differentiable if the marginals of P satisfy some moment assumption, and that gP is strictly concave if the marginals of P are continuous distributions; these remain to be proved, [4].
For certain choices of P and Q, a closed-form expression is available for the shape function. First, we mention how formula (1.8) for the homogeneous models follows from Theorem 2.2. For the exponential model take P " Q " δ c{2 for some c ą 0, and for the geometric model take P " Q " δ ? c for some c P p0, 1q. Then, using (2.6), one can compute that the corresponding shape functions are xy (2.13) for x, y ą 0. Formula (2.12) was first obtained in [13] and formula (2.13) appeared earlier in [3], [7] and [15]. We next list a few models with inhomogeneity for which explicit formulas similar to (2.12) and (2.13) can be derived.
(i) Consider the exponential model. Let c, L, M ą 0 and suppose that P and Q are uniform measures on the intervals rc{2, c{2`Ls and rc{2, c{2`M s, respectively. Here, the left endpoints of the intervals are chosen the same to simplify calculations. Then and Bpzq for z P p´c{2, c{2q. We compute the derivatives as Because A 1 p´c{2q "´8 and B 1 pc{2q "`8, the sectors S1 and S2 are empty. Also, (2.10) becomes Put t " x{y and u " pz`c{2q{p´z`c{2q. Note that u ą 0 and Using the discriminant formula, one finds the solution u ą 0 to be Combining (2.14), (2.15), (2.17) and (2.18) produces the following formula for the shape function The level curve g " 1 is plotted in Figure 2.2 below. For comparison, we included the corresponding level curve of the homogeneous exponential model in which the waiting times have mean m " which is the averaged mean of the waiting times for the exponential model when µ is a product measure. x y p ?
x`?yq 2 " 1{m g " 1 the homogeneous model. This is true more generally and can be seen from the identity xApzq`yBpzq´mp ?
x`?yq 2 " Note from (2.18) that as L, M Ñ 0 one has uptq Ñ ? t, and, thus, the shape function of the homogeneous exponential model mentioned in (2.12). The convergence in (2.21) can also be obtained from the following lemma.
Lemma 2.5. Let a " panq nPN , b " pbnq nPN , c " pcnq nPN and d " pdnq nPN be sequences in D such that an ď cn and bn ď dn for n P N. Suppose that the shape functions exist for both the P a,b -model and the P c,d -model.
(a) If these are exponential models, then g P a,b px, yq ě g P c,d px, yq for all x, y ą 0.
(b) If these are geometric models, then g P a,b px, yq ď g P c,d px, yq for all x, y ą 0.
Proof. We prove (a) only; (b) can be obtained similarly. Fix x, y ą 0. We may assume that g P a,b px, yq ă 8. For ą 0 and N P N, let EN, denote the event The left-hand side is a nondecreasing, continuous function of the waiting times tW pi, jq : i P rM s, j P rN su. Therefore, by the assumption on the parameters and standard properties of stochastic order ([18, Theorem 1.A3(b)]), we have P c,d pEN, q ď P a,b pEN, q.
As N Ñ 8, Gpr N x s, r N y sq{N converges in P a,b -probability to g P a,b px, yq and in P c,d -probability to g P c,d px, yq. Hence, limNÑ8 P a,b pEN, q " 0 and, consequently, limNÑ8 P c,d pEN, q " 0, which implies that g P c,d px, yq ď g P a,b px, yq` . Finally, let Ñ 0.
To see how (2.21) follows, choose the column and the row parameters a " panq nPN and b " pbnq nPN such that the shape function g P a,b exists in model (i). By Lemma 2.5(a), We compute that or z P pc, 1{cq. Again, the sectors S1 and S2 are empty. Now, (2.10) reads which is of the same form as (2.16). Proceeding as in (i), one arrives at the formula gpx, yq " x l logˆ1`L 1´c 2ˆc`1 upxmL{ylM q˙˙`y m logˆ1`M 1´c 2ˆc`u pxmL{ylM q˙ḟ or x, y ą 0 where uptq " 2tp1`M c´c 2 q L´tM`apL´tM q 2`4 tp1`M c´c 2 qp1`Lc´c 2 q for t ą 0.
(iii) Now consider the exponential model choosing P " δ c{2 and Q as the uniform measure on rc{2, c{2`M s, where c, M ą 0 are constants. To identify the shape function, one can repeat the calculation in (i). Alternatively, the shape function can be obtained from (2.19) by letting L Ñ 0, hence, for t ą 0. This can again be justified with Lemma 2.5 as in (i), we omit the details.

The existence and basic properties of the shape function
We will work with the following construction of the P -model. Let us write Ω for R N 2 , the space of all functions ω : N 2 Ñ R`. The set Ω serves as the sample space and each ω P Ω represents a particular realization of the waiting times W . The events are the members of the product σ-algebra. (Each factor R`is equipped with the Borel σ-algebra.) For i, j P N, the waiting time W pi, jq : Ω Ñ R`is the projection map ω Þ Ñ ωpi, jq, and the last-passage time Gpi, jq : Ω Ñ R`is the map defined via formula (1.3). Finally, P , the distribution of the waiting times, is a probability measure on Ω. In particular, we construct the exponential and the geometric models by defining P a,b as the product measure on Ω with the marginals as specified in (1.5) and (1.6), respectively.
The objective in this section is to establish the existence and basic properties of the shape function g P a,b of the P a,b -model for µ-almost every choice of the parameters a and b. For this, we first address the same problem for another corner growth model in which the distribution of the waiting times is an average of P a,b for various choice of a and b. More precisely, define the probability measure P on Ω by for any event B Ă Ω. It turns out that Proposition 3.1. The P-model has a shape function g P . That is, there exists a deterministic function g P : p0, 8q 2 Ñ r0, 8s such that Gpr N x s, r N y sq N " g P px, yq

2)
for all x, y ą 0 almost surely under P. Furthermore, g P is nondecreasing, homogeneous, concave and, if finite, extends continuously to R 2 .
It follows from Proposition 3.1 and (3.1) that for µ-almost every a and b, equality (3.2) holds for all x, y ą 0 almost surely under P a,b as well. Hence, the shape function of the P a,b -model exists and equals g P µ-almost surely.
Since g P is nondecreasing and homogeneous, finiteness of g P at a single point implies that it is finite on p0, 8q 2 . Here homogeneity means that the following identity holds: for any x, y, a ą 0. In fact, in the case of α ă β (see (2.1) and (2.2) for the definitions of α and β), g P is finite. To see this for the exponential model, pick a " panq nPN and b " pbnq nPN such that g P " g P a,b . We may further assume that an ě´α and bn ě β for n P N. By Lemma 2.5 g P a,b is bounded from above by the shape function of the homogeneous P -model, in which the waiting times are distributed exponentially with parameter β´α ą 0. As g P is finite in view of (2.12), g P a,b is finite as well. An alternative means of obtaining finiteness of g P is via inequality (4.10) that appears in the next section.
We next show that P inherits stationarity and ergodicity features of µ. For each pi, jq P Z 2 , define the shift map θi,j : Ω Ñ Ω by θi,jωpk, lq " ωpi`k, j`lq for all ω P Ω and k, l P N.

Proposition 3.2.
(a) P is stationary with respect to θi,j for any i, j P Z`.
(b) P is ergodic with respect to θi,j for any i, j P N.
This allows one to appeal to the subadditive ergodic theory to obtain the existence as well as the concavity of the shape function for the P-model. Stationarity of P also enters the computation of g P in the proof of Proposition 5.2 below. Note that the measure P a,b lacks these properties due to its inhomogeneous structure. Hence, it is more difficult to analyze the shape function of the P a,b -model directly, namely without introducing the averaged P-model. Proof of Proposition 3.2. Let i, j P Z`. For sequences a " panq nPN and b " pbnq nPN in D, the marginal of P a,b on pi, jqth coordinate depends only on the terms ai and bj. It follows from this that P a,b pθ´1 i,j pBqq " P τ i a,τ j b pBq (3.3) for any event B Ă Ω, where τ k denotes the shift map pcnq nPN Þ Ñ pc n`k q nPN on D N . Integrating (3.3) against µ over D NˆDN and using stationarity of µ with respect to τiˆτj yield Ppθ´1 i,j pBqq " PpBq for any event B Ă Ω, which is (a). Suppose now that i, j P N and θ´1 i,j pBq " B. Let T n denote the σ-algebra generated by the collection of random variables An " tW pk, lq : ipn´1q ă k and jpn´1q ă lu for n ě 1. Then B is in Ş nPN T n, which is the tail σ-algebra associated with the the sequence of σ-algebras generated by An An`1 since i, j ą 0. Because P a,b is a product measure, by Kolmogorov's 0´1 law, P a,b pBq P t0, 1u for any sequences a and b in D. Therefore, PpBq " µpP a,b pBq " 1q. On the other hand, by virtue of (3.3), Since µ is ergodic with respect to τiˆτj, we conclude that PpBq P t0, 1u.
The assertions made in Proposition 3.1 except for the finiteness of the shape function are proved in [16, Chapter 2] for homogeneous P -models, that is, when P is i.i.d. As noted below, the proof goes through if P is, more generally, ergodic with respect to θi,j for all i, j P N. We remark that the special structure of P given by (3.1) is not used in the proof. where the maximum is taken over all directed paths from pK, Lq to pM, N q. In particular, GpM, N q " Gpp1, 1q, pM, N qq. Fix x, y P N and define the random variables ZpM, N q "´GppM x`1, M y`1q, pN x, N yqq for 0 ď M ă N . It follows from definition (3.4) and Proposition 3.2 that pZpM, N qq0ďMăN is a subadditive process that satisfies the hypotheses of the Liggett's version of subadditive ergodic theorem in [8]. Hence, Zp0, N q{N " Gpr N x s, r N y sq{N converges P-almost surely to a deterministic limit, g P px, yq.
To establish the existence of the limit in (3.2) for all x, y ą 0 P-almost surely, and that g P is nondecreasing, homogeneous and concave, we use Proposition 3.2 again together with monotonicity and superadditivity of the last-passage times. These properties are immediate from nonnegativity of the waiting times and (3.4). We omit the details of the argument, which are the same as in the i.i.d. case and can be found, for example, within the proof of Theorem 2.1 in [16].
Finally, we show that g P extends continuously to R 2 if g P px, yq is finite for all x, y ą 0. As a finite concave function on p0, 8q 2 , g P is continuous on p0, 8q 2 . Extend g P to R 2 by setting g P p0, 0q " 0 g P px, 0q " lim yÑ0 g P px, yq for x ą 0 g P p0, yq " lim xÑ0 g P px, yq for y ą 0, where both limits exist because g P is nondecreasing. To verify that g P is continuous on the x-axis, for example, pick x0 ą 0, δ P p0, x0q and let R δ denote the rectangle tpx, yq : x P rx0´δ, x0`δs and y P p0, δqu. Then, since g P is nondecreasing and homogeneous, Ñ 0 as δ Ñ 0.

A family of stationary corner growth models on Z 2
The approach taken in this paper to identify the shape function of the P-model is based on consideration of certain corner growth models on Z 2 that are stationary in a sense made precise below. The useful feature of these models is that their shape functions are explicit and related to g P through a variational formula. This approach is exploited in [16,Chapter 2] to study the homogeneous geometric model. Their arguments can be generalized to treat the inhomogeneous exponential and geometric models as well; we present the details in this section. Systematic use of stationary processes to calculate explicit limit shapes first appeared in [15].
In general, similar to the construction of the models on N 2 , we identify a corner growth model on Z 2 with a probability measure P on the space p Ω of all functions ω : Z 2 Ñ R`, and we refer to it as the P -model. The σ-algebra on p Ω " R Z 2 is the product of the Borel σ-algebras on each factor R`. The measure P represents the distribution of the waiting times tW pi, jq : i, j P Z`u. The last-passage times are defined by p GpM, N q " max π ÿ pi,jqPπ W pi, jq (4.1) for M, N P Z`, where the maximum is taken over all directed paths from p0, 0q to pM, N q. We remark that p G obeys recursion (1.1) with boundary conditions p Gpi, 0q " ÿ kPris W pk, 0q and p Gp0, jq " ÿ kPrjs W p0, kq for i, j P N .
Finally, we define the shape function p gP of the P -model as Gpr N x s, r N y sq N provided that the limit exists and is deterministic for all x, y ě 0 almost surely under P . Let π denote the projection map p Ω Ñ Ω that sends p ω P p Ω to ω P Ω defined by ωpi, jq " p ωpi, jq for all i, j P N. Any probability measure p P on p Ω induces a probability measure P on Ω defined by P pBq " p P pπ´1pBqq for any event B Ă Ω. We refer to P as the projection of p P onto Ω. We will identify any random variable X : Ω Ñ R with the random variable X˝π : p Ω Ñ R. In particular, we may think of the random variables W pi, jq and Gpi, jq as defined on p Ω. Fix sequences a " panq nPN and b " pbnq nPN in D, and consider the P a,b -model. We first work in the exponential setting, that is, the marginals of P a,b are given by (1.5). Recall the definition of α and β in (2.1). Suppose that an ě´α and bn ě β for all n P N. This assumption holds µ-almost surely.
For each z P pα, βq, we now construct a corner growth model on Z 2 from the P a,b -model by introducing waiting times at the boundary sites pi, 0q, p0, jq for i, j P Z`with suitable distributions. Define the probability measure P z a,b as a product measure on p Ω by specifying that P z a,b pW p0, 0q " 0q " 1 P z a,b pW pi, 0q ě xq " expp´pai`zqxq, P z a,b pW p0, jq ě xq " expp´pbj´zqxq, P z a,b pW pi, jq ě xq " expp´pai`bjqxq, for x ě 0 and i, j P N. In particular, the projection of P z a,b onto Ω is P a,b . Note that since z P pα, βq both ai`z and bj´z, being positive, are valid parameters for the exponential distribution. We call this model an exponential model.
We also describe the analogous construction for the geometric setting. Now, α and β are as in (2.2), the marginals of P a,b are given by (1.6), and we suppose that an ď α and bn ď 1{β for all n P N. For each z P pα, βq, define the probability measure P z a,b as a product measure on p Ω with marginals P z a,b pW p0, 0q " 0q " 1 P z a,b pW pi, 0q " kq " p1´ai{zqpai{zq k P z a,b pW p0, jq " kq " p1´bjzqpbjzq k P z a,b pW pi, jq " kq " p1´aibjqpaibjq k for k P Z`and i, j P N. Again, P a,b is the projection of P z a,b onto Ω. Also, both ai{z and bjz are valid parameters for the geometric distribution as they lie in the interval p0, 1q. We call this model a geometric model.
A remarkable aspect of the P z a,b -model is that, the horizontal and vertical increments of the last passage times are stationary. To make a more precise statement, define the increment variables as  Using (1.1), one can develop the following recursion ((2.21) in [16]) for the increment variables.   The proof of Proposition 4.1 is similar to that of Theorem 2.4 in [16]. We utilize the following lemma.
Lemma 4.2. Let F : R 3 Ñ R 3 denote the mapping px, y, zq Þ Ñ px´x^y`z, y´x^y`z, x^yq Let P denote a product probability measure on R 3 with marginals P1, P2, P3. Suppose that one of the following conditions holds: (i) P1, P2 and P3 are exponential distributions with paramaters a, b and a`b, for some a, b P p0, 8q.
(ii) P1, P2 and P3 are geometric distributions with (fail) parameters a, b and ab, for some a, b P p0, 1q.
Then F preserves P , that is, P pF´1pBqq " P pBq (4.7) for any Borel set B in R 3 .
Case (ii) is stated as Lemma 2.3 in [16]. The other case partially features in some form as Lemma 4.1 in [1]. For proof, one compares the Laplace transforms of measures P and P pF´1p¨qq. We include another proof below.
Proof of Lemma 4.2. Observe that F is a bijection on R 3 with F´1 " F . Let us consider (i), first. It suffices to verify (4.7) for any open set B in R 3 . Note that F is continuous and, thus, F´1pBq is also an open set. Furthermore, F is differentiable on the open set tpx, y, zq : x ą y or x ă yu with absolute value of its Jacobian equal to 1. Hence, we can appeal to the change of variables theorem, [14,Theorem 7.26], which gives P pF´1pBqq " abpa`bq ż F´1pBq e´a x´by´pa`bqz dxdydz " abpa`bq ż F´1pBq e´a px´x^y`zq´bpy´x^y`zq´pa`bqpx^yq dxdydz " abpa`bq ż B e´a u´bv´pa`bqw dudvdw " P pBq.
The proof of (ii) is a discrete version of the preceding argument. Now, it suffices to verify (4.7) with B Ă Z 3 . Note that F´1pBq " F pBq Ă Z 3 as well. Hence, Proof of Proposition 4.1. For convenience of presentation, let us assume that the model is exponential; the argument proceeds similarly for geometric models. We do induction on k`l. The base case k " l " 0 holds by (4.2) because Ipi, 0q " W pi, 0q and Jp0, jq " W p0, jq for i, j P N. Now suppose that k`l ą 0, and (a)-(c) are true with k`l´1. Without loss of generality, l ą 1. To complete the induction argument, it suffices to verify the following: For any m P Z`with m ě k (i) The collection of random variables tJpk, jq : j ą lu Y tIpi, lq : k ă i ď mu Y tJpm, lqu Y tIpi, l´1q : i ą mu are independent.
(ii) Jpm, lq is distributed exponentially with parameter b l´z .
(iii) If m ą k, Ipm, lq is distributed exponentially with parameter am`z.
We argue by induction on m. For m " k, (i) and (ii) follow from the inductive assumption on k, l´1, and (iii) is vacuously true. Suppose that m ą k, and (i)-(iii) hold for m´1. Note that by (4.5) and (4.6) pIpm, lq, Jpm, lq, Ipm, l´1q^Jpm´1, lqq " F pIpm, l´1q, Jpm´1, lq, W pm, lqq (4.8) where F is as defined in Lemma 4.2. Furthermore, Ipm, l´1q, Jpm´1, lq and W pm, lq are independent, and are all exponentially distributed with parameters am`z, b l´z and am`b l , respectively. Here, we rely on (i), (ii) with m´1 and (a) with l´1, and that Ipi, jq and Jpi, jq depend only on the waiting times W pu, vq with u ď i and v ď j. Now, (ii) and (iii) are immediate from an application of Lemma 4.2. Also, because the random variables on the right-hand side of (4.8) are independent of tJpk, jq : j ą lu Y tIpi, lq : k ă i ă mu Y tIpi, l´1q : i ą mu (i) follows.
By virtue of Proposition 4.1, one can compute the shape function for the P z a,b -model.  Let gz : R 2 Ñ R`denote the function px, yq Þ Ñ xApzq`yBpzq. Note the inequality gpx, yq ď gzpx, yq (4.10) for any x, y ě 0. This follows from the inequality GpM, N q ď p GpM, N q, which is immediate from a comparison of the definitions (1.3) and (4.1), and that P a,b is the projection of P z a,b onto Ω. In section 3, we justified the finiteness of g P via Lemma 2.5. Now this also follows from Theorem 4.3 and the inequality (4.10).
As in section 3, one can also consider an averaged corner growth model on Z 2 for each z P pα, βq, where the distribution of waiting times is defined by for any event B Ă p Ω. Statement (4.9) holds P z -almost surely as well, since gz does not depend on the parameters a and b. In other words, gz is also the shape function for the P z -model. Also, note that the projection of P z onto Ω is P defined in (3.1).
We now point to a connection between (1.7) and a result from [4]. For m, n, k, l P Z`, define Cppm, nq, pk, lqq " p Gpm, nq´p Gpk, lq. Hence, C is a map from p ΩˆZ 2ˆZ2 into R. The following properties hold for any p, q P Z`and z P pα, βq.
(iii) Cppm, nq, pk, lqq has the same distribution as Cppm`p, n`qq, pk`p, l`qqq under P z .

(i) is clear and (ii) follows because p
Gpm, nq is bounded by the sum of W pi, jq with 0 ď i ď m and 0 ď j ď n. In view of (4.4), (4.5) and (4.6), Cppm, nq, pk, lqq is a certain measurable function F of the increment variables Ipi, n^lq and Jpm^k, jq with m^k ă i ď m _ k and n^l ă j ď n _ l. Note that F depends on m, n, k, l only through the numbers m _ k´m^k and n _ l´n^l. Using this, (iii) can be obtained from Proposition 4.1 and ergodicity of µ with respect to the shift τpˆτq.
Due to the similarity of properties (i)-(iii) to the ones listed in [4, Definition 2.1], it seems appropriate to call C a stationary, P z -integrable cocycle for each z P pα, βq. Note also that Apzq and Bpzq are the expectations under P z of Cpp0, 0q, p1, 0qq and Cpp0, 0q, p0, 1qq, respectively. Therefore, (1.7) can be viewed as a variational formula over a collection of cocycle distributions akin to [4, (2.16)]. Moreover, as noted in the proof of Corollary 2.4, if px, yq P S then there exists z P pα, βq such that gpx, yq " xApzq`yBpzq. This is formally in agreement with the formulas in [4,Theorem 4.6], which are only claimed for homogeneous models.
The proof of Theorem 4.3 is adapted from that of Corollary 2.5 in [16].
Proof of Theorem 4.3. The statement (4.9) is clearly true if x " 0 and y " 0. Without loss of generality, we may assume that x ą 0. From (4.4), we obtain that Jpr N x s, jq.
We will show that, P z a,b -almost surely, Jpr N x s, jq Ñ yBpzq (4.13) as N Ñ 8. There is nothing to prove in (4.13) if y " 0; hence, assume that y ą 0 as well.
We will consider the exponential model only; the proof for the geometric model is similar. By Proposition 4.1, under P z a,b , Ipi, 0q is exponentially distributed with parameter ai`z for i P N, and the random variables tIpi, 0q : i P Nu are independent. In addition, Jpr N x s, jq is exponentially distributed with parameter bj´z for j P N and the random variables tJpr N x s, jq : j P Nu are independent. Let ą 0. Applying a large deviation estimate to be justified below, we get p´α`zq x`1^1˙2˙( 4.14) Jpr N x s, jq´1 bj´zˇˇˇˇą Since the bounds in (4.14) and (4.15) are summable over n P N, by the Borel-Cantelli lemma, as N Ñ 8 almost surely under P z a,b . On the other hand, by the pointwise ergodic theorem, for µ-almost every a and b, we have Qpdbq " yBpzq. Combining (4.16) and (4.17) gives (4.12) and (4.13). It remains justify the estimates (4.14) and (4.15). Let n P N and, for i P rns, let Xi be a random variable defined on Ω, exponentially distributed with parameter ci ą 0 under the measure P z a,b . Suppose also that tXi : i P rnsu are independent. Let 0 ă t ă c{2 where c " min iPrns ci. Using Markov's inequality and independence, we obtain that  For the second inequality above, we use that Finally, we observe that (4.20) implies (4.14) and (4.15).

Variational characterization of the shape function
The purpose of this section is to obtain the proofs of Theorem 2.2 and 2.3. As noted after Proposition 3.1, for µ-almost every choice of the column and the row parameter a and b, the shape function of the P a,b -model exists and is identical to the shape function of the P-model. The remaining task is to derive (2.6) for the case α ă β, and (2.7) and (2.8) for the case α " β We focus on (2.6) first. Let us write g for the shape function g P as well as for the shape function g P a,b for a typical choice of a and b for which g P " g P a,b ; this causes no ambiguity and simplifies the notation. Also recall from section 4 that gz is the shape function of the P z a,b -model. We begin with computing the shape function g on the boundary. By homogeneity of g, it suffices to compute gp1, 0q and gp0, 1q. Recall from Proposition 3.1 that g is extended to the boundary of R 2 by continuity.
To obtain the lower bound, we begin with the inequality where ą 0. In the rest of the argument, we work with the exponential model; the geometric model can be treated similarly. For µ-almost every a " panq and b " pbnq, P a,b -almost surely. Let δ ą 0. By the pointwise ergodic theorem, µ-almost surely. Now fix a " panq and b " pbnq such that (5.2) and (5.3) are in force and b1 ď β`δ. By (5.1) and the condition on b1, ai`b1˙.
Since tW pi, 1qu iPN are independent exponential random variables with parameters bounded below by b1 ą 0, we can apply the large deviation estimate (4.20) and the Borel-Cantelli lemma to conclude that, P a,b -almost surely, the second term on the right-hand side vanishes as N Ñ`8. Combining this with (5.2) and (5.3), we obtain gp1, q ě Apβ`δq. Letting , δ Ñ 0 gives the lower bound. Computation of gp0, 1q is similar to that of gp1, 0q, which we omit.
Next we relate the shape function g to the shape functions of the family of the stationary models introduced in section 4.
(5.4) Proposition 5.2 is stated (as Proposition 2.7) and proved in [16] for the homogeneous geometric model. In the proof below, we adapt the argument there for the ď half of (5.4).
By ( Also, because G and p G are both nondecreasing, it follows from (5.5) that where the maximum is now taken over a finite set that does not depend on N . By Proposition 3.2, one has the following equalities in distribution under P.
in P-probability as N Ñ 8 since the limits are deterministic. Hence, these limits are P-almost sure if N tends to infinity along suitable subsequences. Divide through by N in (5.6), let N Ñ 8 along suitable subsequences and consider the limit of each term under P z defined in (4.11). By Theorem 4.3 and (5.7) Finally, let L Ñ 8.
Now, we set out to extract g from (5.4). For this, we will use the boundary values of g provided in Lemma 5.1, and that A and B are continuous stricly monotone functions on pα, βq. At this point, the details of the corner growth model will not play a role. The following lemma, which is a consequence of Proposition 5.2, simplifies the task at hand. Note that (5.8) follows because θ Þ Ñ px, yq is a polar curve that does not pass through the origin. In particular, x and y are bounded away from 0, respectively, on the intervals r0, π{4s and rπ{4, π{2s.
Collecting the terms on the right-hand side of (5.8) and using the homogeneity of g, one obtains that The expressions inside the supremums in (5.9) are continuous functions of θ over closed intervals. Hence, there exists θz P r0, π{2s such that where the second equality is due to (4.10). This equality remains valid if T is replaced with`8 in view of (5.12) with y " 1. Rearranging terms gives (5.10).
Let us introduce the function γ : r0, 8q Ñ R as γptq "´gpt, 1q. It follows from Proposition 3.1 that γ is nonincreasing, continuous and convex on r0, 8q and completely determines g. Let us also denote with f the function whose graph is the image of the curve z Þ Ñ p´Apzq, Bpzqq.
That is, f is defined on the interval p´Apαq,´Apβqq and is given by the formula f pxq " B˝A´1p´xq. By Corollary 5.4, f pxq " sup 0ďtă8 ttx´γptqu (5.13) for any x P p´Apαq,´Apβqq. Note that (5.13) is a convex duality statement, which can be utilized to give a proof of Theorem 2.2.
Since γ is a lower semi-continuous, proper convex function on the real line, by the Fenchel-Moreau theorem, γ equals the convex conjugate of γ˚, hence, γptq " sup xPR ttx´γ˚pxqu (5.15) for t P R.
Our aim is to show that the supremum in (5.15) could be taken over the interval p´Apαq,´Apβqq instead of the real line. For this, we need to examine the values of γ˚on the complement of p´Apαq,´Apβqq. It is clear from (5.14) that γ˚is nondecreasing and is bounded below bý γp0q " gp0, 1q " Bpαq.
Finally, we compute γ˚at´Apβq. Being a convex conjugate, γ˚is lower semi-continuous.
The conclusion is that the function x Þ Ñ tx´γ˚pxq is increasing for x ď´Apαq and is´8 for x ą´Apβq. Moreover, the left-and right-hand limits agree with the value of the function at´Apβq and´Apαq, respectively. Hence, by (5.15), Finally, we complete the proof of Theorem 2.3.
Proof of Theorem 2.3. As usual we prove (a); the same argument with minor modifications yields (b) as well.
Choose the column and the row parameters a " panq nPN and b " pbnq nPN such that the P a,b -model has a shape function g. Note that definition (4.2) of the probability measure P 0 a,b makes sense even under the current assumption α " β " 0. Using inequality (4.10) and Theorem 4.3, we obtain the upper bound gpx, yq ď p g0px, yq " xAp0q`yBp0q for x, y ą 0, (5.17) where Ap0q and Bp0q refer to the integrals in (2.3) with z " 0. The right-hand side of (5.17) may be infinite. Now the lower bound. For c ą 0, let us write Φc for the map pcnq nPN Þ Ñ pcn _ cq nPN defined on D N . Note that for any i, j P Z`, the induced measure µc of µ under the map ΦcˆΦc also satisfies the stationarity and ergodicity assumptions ((i) and (ii) on p.4) because Φc commutes with τ k for any k P Z`. Therefore, we can apply Theorem 2.2 with µc. Let Pc and Q c denote the marginal distributions of the projections of µc onto the first and the second coordinates, respectively. We may assume that the P pan_cqn,pbn_cqn -model has a shape function gc as well. Then, by Theorem 2.2, gcpx, yq " inf p´c,cq txAcpzq`yBcpzqu, where Ac and Bc are defined as in (2.3) but with Pc and Q c replacing P and Q, respectively. That is, Qpdbq.