Scaling limits for non-intersecting polymers and Whittaker measures

We study the partition functions associated with non-intersecting polymers in a random environment. By considering paths in series and in parallel, the partition functions carry natural notions of subadditivity, allowing the effective study of their asymptotics. For a certain choice of random environment, the geometric RSK correspondence provides an explicit representation of the partition functions in terms of a stochastic interface. Formally this leads to a variational description of the macroscopic behaviour of the interface and hence the free energy of the associated non-intersecting polymer model. At zero temperature we relate this variational description to the Marchenko-Pastur distribution, and give a new derivation of the surface tension of the bead model.


Introduction and summary
We study the partition functions associated with a natural model for non-intersecting polymers in a random environment. Apart from being an interesting physical model in its own right, this model is motivated by recent developments on connections between random polymers and Whittaker functions, obtained via the geometric RSK correspondence [11,31]. By considering paths in series and in parallel, the partition functions carry natural notions of subadditivity allowing the effective study of their asymptotics. In particular we show that, for a small number of paths, the free energy has a linear dependence on the number of paths (Theorem 2.2). Interestingly, it seems quite difficult to prove this using the integrable structure of the log-gamma polymer model; instead we give a general argument, valid for general weights. We also determine the correct order of scaling for the partition function associated with a large number of paths (see (2.14), (2.15) and Corollary 2.7).
We then turn our discussion to the random polymer model with log-gamma weights, where the geometric RSK correspondence provides an explicit representation of the partition functions in terms of a stochastic interface [11,15]. Formally this leads to a variational description of the macroscopic behaviour of the interface and hence the free energy of the associated non-intersecting polymer model. At zero temperature we relate this variational description to the Marčenko-Pastur distribution, and give a new derivation of the surface tension of the bead model, which was recently computed by Sun [41] using very different methods. In the following, we summarise the main results of the paper, beginning with some preliminary definitions.

Preliminary definitions
Let e 1 = (1, 0) and e 2 = (0, 1) be the standard basis for Z 2 . For points x and y in Z 2 , a path π from x to y is a set of points π = {a 0 , . . . , a p } in Z 2 such that a 0 = x, a p = y, and a j+1 − a j is equal to either e 1 or e 2 for every 1 ≤ j < p.
We say a vector x = (x 1 , . . . , x k ) ∈ Z 2×k is a k-point if x 1 , . . . , x k are distinct points in Z 2 . Suppose x = (x 1 , . . . , x k ) and y = (y 1 , . . . , y k ) are k-points and for each i, π i is a path from x i to y i . If the sets π 1 , . . . , π k are disjoint, we say the k-tuple π = (π 1 , . . . , π k ) is non-intersecting and refer to π as a k-path. (We emphasise that k-paths will always be non-intersecting.) We write Γ x→y for the set of k-paths from x to y, and write x ≤ y whenever Γ x→y is non-empty.
Suppose further we have a random environment {ω(z) : z ∈ Z 2 }-a collection of independent and identically distributed finite-expectation random variables under a probability measure P. For a k-path π = (π 1 , . . . , π k ) from x to y, we define the energy F (π) of π to be the sum of the weights included in the k-path, not including the starting points. That is, F (π) is the random variable associated with the non-intersecting directed random polymer running from x to y, with a particular emphasis on the asymptotics of Z x→y (β) as the lengths of the paths and/or the number of paths grows to infinity. We emphasise that the assumptions E[ω(0, 0)] < ∞ and G(β) = E[e βω(0,0) ] < ∞ are in force throughout the paper.
This article is divided into three interconnected parts. The first part is dedicated to proving asymptotic results for the partition functions in a general environment. In the second part we introduce a collection of random functions on the N × N square known as stochastic interfaces, and express the non-intersecting partition functions associated with a particular random environment in terms of a stochastic interface, developing a relationship with random matrix theory in the process. In the final part, we study the scaling limits of these interfaces as N → ∞, relating our model to the Marčenko-Pastur law and Gaussian fields, and developing the machinery along the way to obtain new derivation of the surface tension of the bead model. In the remainder of Section 1 we overview each of these three respective parts in more detail.
Before proceeding, we mention a couple of notational conventions we use to lighten notation. Whenever (a n ) n∈Z are variables indexed by the integers, for real numbers r we set a r := a ⌊r⌋ , where ⌊r⌋ is the largest integer not greater than r. Similarly, if a is a variable indexed by Z 2 then for real numbers r and s, a(r, s) will refer to a(⌊r⌋, ⌊s⌋). If f is a function defined on a subset of Z 2 or R 2 , we say f is symmetric if f (r, s) = f (s, r) for every s and r. Finally, if x is a k-point and a is an element of Z 2 , define the translation x + a to be the k-point whose i th component is given by x i + a.

Non-intersecting polymers
The first part of this article is concerned with studying key properties of non-intersecting polymers in a general environment, with a particular focus on developing tools to study their asymptotics. These results are stated in full in Section 2, though we provide a brief summary here.
We say a k-point x = (x 1 , . . . , x k ) is nice if either x i+1 lies strictly north and strictly west of x i , or x i+1 = x i + e 2 . This is a technical condition which roughly speaking ensures that k-paths going to and from a k-point x do not intersect. (See the beginning of Section 2 for a more precise statement as well as a proof.) Our main tools for tackling the asymptotics of non-intersecting polymer partition functions Z x→y (β) are the dual notions of series and parallel concatenation, which we now outline. Given a k-path π from x to y and k-path γ from y to z, we may concatenate the paths π and γ in series to form a new k-tuple π ⊕ γ of paths from from x to z. Provided the intermediate k-point y is nice, the concatenated path π ⊕ γ turns out to also be non-intersecting, and in this case the operation ⊕ is additive in the sense that F (π ⊕ γ) = F (π) + F (γ).
In particular, we have an F -preserving injection taking every pair of paths in Γ x→y × Γ y→z to a new path π ⊕ γ in Γ x→z . In Section 2, we use this injection to prove the series bound, which states that for nice k-points x ≤ y ≤ z, the partition functions satsify Z x→z (β) ≥ Z x→y (β)Z y→z (β). (1. 3) The series bound gives rise to subadditivity in the logarithmic partition functions log Z x→y (β), which we use in conjunction with Kingman's subadditive ergodic theorem [22] to obtain the logarthimic asymptotics of Z x→y (β) as x and y grow far apart in the (1, c) direction. Roughly speaking, Theorem 2.1 and Theorem 2.2 state that for any pair of nice k-points x and y 1 N log Z x→y+(N,cN ) (β) converges to a deterministic limit kf c (β), (1.4) where the constant f c (β) is independent of x, y and the dimension k.
The parallel bound (1.5) gives a subadditivity in the k-variable which allows us to prove a result analogous to (1.4) concerning infinitely many paths of finite length running side-by-side. Thereafter, we study the infinite temperature (β = 0) case, where the Lindström-Gessel-Viennot formula may be used in conjunction with Szegö's powerful limit theorem to obtain explicit expressions for the large-k logarithmic asymptotics of the partition funcitions in terms of a Laurent series with binomial coefficients.
Finally we look at the case of many long paths. By studying the large-N asymptotics of a variant of Macmahon's formula [40], in Lemma 2.5 we obtain explicit 1 N 2 logarithmic asymptotics for the partition functions associated with the infinite temperature case. We then use Jensen's inequality to relate the positive temperature partition functions with their infinite temperature counterparts (Lemma 2.6), leading us to Corollary 2.7, which establishes N 2 behaviour for the logarithmic partition function when both the number of paths aswell as their lengths are of order N .

Stochastic interfaces and Whittaker measures
In the second part of this article-the results of which we state in full in Section 3-we introduce stochastic interfaces, random functions defined on the square S N := {1, . . . , N } 2 of Z 2 . For a convex interaction potential V , a stochastic interface is essentially a random function φ N : S N → R whose law is proportional to 6) where D N := {(i, i) : i = 1, . . . , N } is the diagonal of S N , and x, y is the set of edges in S N directed in the north or east direction. The central idea connecting directed polymers with stochastic interfaces is Proposition 3.2, which states that the non-intersecting partition functions associated with a certain random environment may be jointly expressed in terms of a stochastic interface.
More specifically, let τ N µ (m, k) be the partition function associated with a k-paths in an m × N rectangle, where the variables in the random environment are log-gamma distributed in that each e βω(z) has the inverse gamma law of parameter µ > 0. We now define a random function ϕ N µ : S N → R in terms of the joint partition functions τ N µ (m, k) by setting for i ≤ j (with a similar definition holding for i ≥ j). Using a restatement of a theorem by Corwin, O'Connell, Seppäläinen and Zygouras [11], we obtain Proposition 3.2, which states that ϕ N µ is a stochastic interface with interaction potential V (u) = exp(u).
The small-µ and large-µ deformations of the stochastic interface ϕ N µ have respective connections with the eigenvalues ensembles of random matrix theory and the partition functions of deterministic nonintersecting directed polymers. In the former case, by appealing to ideas in Baryshnikov [3], we show that an eigenvalue process ϕ N LUE associated with the minors of a random matrix from the Laguerre Unitary Ensemble may be expressed in terms of a stochastic interface with the hard-core interaction potential V (u) := ∞½ u>0 . We use the results of [11] to obtain Proposition 3.4, which states that µϕ N µ converges in distribution to ϕ N LUE as µ → 0. (1.8) Using a tropicalisation of our underlying polymer, the statement (1.8) recovers a result by Johannson [18] relating non-intersecting last passage percolation with exponential weights to the eigenvalues of a Laguerre matrix.
We have contrasting behaviour as µ → ∞. Namely, if we define the tilted interface by θ N µ (i, j) := ϕ N µ (i, j) + (2N + 1 − (j + i)) log µ, then according to Theorem 3.6 we have θ N µ converges almost surely to a deterministic function θ N min as µ → ∞. (1.9) Moreover, the deterministic limit θ N min is the minimiser of a discrete variational problem on the square S N , and has an explicit representation in terms of factorials.

Scaling limits for stochastic interfaces
The final part of this article, which is discussed in full in Section 4, is concerned with using variational heuristics to study the large-N asymptotics of stochastic interfaces. In the interest of maintaining a steady flow of ideas, in this part of the article we will be content to provide a plausibility argument based on physical heuristics in place of a rigorous mathematical proof.
Following the exposition in Funaki and Spohn [16], our first tool for studying the macroscopics of stochastic interfaces is that of surface tension-an asymptotic measure of the energy cost for an interface to lie a certain tilt. With this concept at hand, we conjecture that for certain choices of interaction potential, the macroscopic shapes of certain square interfaces may be given as solutions to variational problems on the unit square S := [0, 1] 2 . In particular we anticipate that ifφ N µ : S → R is a rescaling of the interface associated with the partition functions of the non-intersecting log-gamma polymer-with both the square S N and the height of the interface both rescaled by 1 N -then ϕ N µ converges to a deterministic limit ξ µ : S → R as N → ∞.
Moreover, if we let σ exp be the surface tension associated with the exponential interaction potential V (u) = exp(u), then we expect that the deterministic limit ξ µ is the the minimiser of where the minimisation is taken over all functions v : S → R satisfying v(1, 1) = 0. Finally, the limit shape is related to the asymptotic partition functions of the non-intersecting directed log-gamma polymer through the following equation. Let τ N µ (m, k) be the partition function associated with k paths on an N × m rectangle. Then these arguments suggest that Having taken N → ∞, we now consider the behaviour of ξ µ as the parameter µ varies. On the one hand, by considering the relationship the interface ξ N µ has with the eigenvalues of Laguerre matrices, taking a large-N analogue of (1.8) we anticipate that µξ µ converges to ξ mp as µ → 0, where ξ mp has an explicit expression in terms of the Marčenko-Pastur distribution. On the other hand, in the large-µ case we sketch an argument suggesting that where ξ ht (s, t) has an explicit expression in terms of the function q(u) = u log u. Given their own direct representations as scaling limits of stochastic interfaces, we also expect both limit functions ξ mp and ξ ht to be solutions of explicit variational problems.
Thereafter we take a more refined argument using the central limit theorem, studying the case where µ and N are sent to ∞ together through the scaling limit µ = κN 2 . Indeed, Conjecture 4.7 states that the random processes H N κ := {H N κ (s, t) : 0 ≤ s ≤ t ≤ 1} given by converge in distribution to a centred Gaussian process, and gives a prediction for the covariance functions in terms of the asymptotic densities of a path model.
Finally, having developed the surface tension framework to study the macroscopics of stochastic interfaces, we use a version of the classical semicircle law of random matrix theory to reverse-engineer a straightforward derivation of a formula for the surface tension associated with the bead model [4]. To our knowledge, the only other place this formula appears in the literature is in the work of Sun [41], who provides an advanced derivation involving scaling limits of dimer models [8].

Outline of the paper
The remainder of the paper is structured as follows. In Section 2, we discuss the asymptotics of nonintersecting directed polymers in random environments with a general weight distribution, giving full statements of the results overviewed in Section 1.2. In Section 3, we introduce stochastic interfaces and their connections with random matrices, fleshing out the description seen in Section 1.3. Finally, in Section 4, we study the large-N asymptotics of the interfaces seen in Section 3, expanding on the discussion in Section 1.4.
The remaining sections of the paper, Sections 5-8, are dedicated to giving proofs of results stated in Section 1 to Section 4, as well as providing further details in support of predictions made in Section 4. Section 5 and Section 6 are dedicated to proving the results stated in Section 2 and Section 3 respectively. The proofs and further details surrounding the discussion in Section 4 are spread across two sections, the latter of which, Section 8, is dedicated to our derivation of the surface tension of the bead model. 2 Non-intersecting polymers

Series and parallel inequalities
In this section we provide brief proofs of the series and parallel bounds. First we consider the series bound (1.3), which we prove using concatenation in series as follows. Suppose x, y, z are points in Z 2 , and π = {a 0 , . . . , a p } is a 1-path from x to y, and γ = {b 0 , . . . , b q } is a 1-path from y to z. Then we define the concatenated 1-path π ⊕ γ := {c 0 , . . . , c p , c p+1 , . . . , c p+q } from x to z by setting c i = a i for i ≤ p and It is plain to check that this is indeed a path from x to z, and that Now suppose x, y and z are k-points satisfying x ≤ y ≤ z. If π is a k-path from x to y and γ is a k-path from y to z, then we may form a k-tuple π ⊕ γ := (π 1 ⊕ γ 1 , . . . , π k ⊕ γ k ) of paths from x to z, which may or may not be intersecting. Recall that we say a k-point (x 1 , . . . , x k ) is nice if either x i+1 lies both strictly north and strictly west of x i , or x i+1 = x i + e 2 . We now prove the series bound (1.3), which hinges on the observation that whenever the intermediate point y is nice, then this resulting k-tuple (π ⊕ γ) is guaranteed to be non-intersecting.
Proof of equation (1.3). First we show that if y is nice, then π ⊕ γ is a k-path-that is the sets (π i ⊕ γ i ) and (π j ⊕ γ j ) are disjoint for each i = j. Since each of π and γ are themselves k-paths, it is sufficient to show that for each i = j, the sets π i and γ j are disjoint.
To see this, first consider the case i < j. In this case y j lies strictly north of y i , and hence every point of γ j lies strictly to the north of every point of π i , and hence π i ∩ γ j = ∅. Alternatively, consider the case i > j. Since y is nice, either y i lies both strictly north and strictly west of y j , or y i is of the form y i = y j + me 2 for some positive integer m. In the former case, every point of γ j lies strictly east of every point of π i , and hence π i ∩ γ j = ∅. In the latter case, we must have y j+1 = y j + 1e 2 , and in particular, γ j must go through y j and then y j + 1e 1 in order avoid y j+1 = y j + 1e 2 . It follows that γ j − {y j } consists of points that lie strictly east of π i , and hence again π i ∩ γ j = ∅.
We have proved that all cases π i and γ j are disjoint, and hence the k-tuple π ⊕ γ of paths from x to z is a k-path. In particular, ⊕ is an injection from Γ x→y to Γ y→z → Γ x→z . Observing that the paths π and γ only overlap at y, we see that this operation is additive in the sense that Now suppose we have three k-points x ≤ y ≤ z such that y is nice. Using the positivity of the summands in (1.2) with the fact that ⊕ is an injection to obtain the second line below, and (2.1) to obtain the third, We now consider the dual procedure of concatenation in parallel, giving a proof of the parallel bound (1.5).
We remark that by iterating (1.5), for any pair of k-points x = (x 1 , . . . , x k ) and y = (y 1 , . . . , y k ) we obtain In fact, thanks to the celebrated Lindström-Gessel-Viennot lemma [40], the k-path partition functions have a determinantal expression in terms of the one-point partition functions: In light of (2.4), the inequality (2.3) may be understood as an analogue of Hadamard's inequality for the matrix (Z xi→yj ) k i,j=1 .

The single path partition function
The case k = 1 corresponds to studying the asymptotics of the partition function Z (1,1)→(n,m) (β) as m and n become large, and is widely discussed in the literature. By a standard subadditivity argument using the special case k = 1 of the series bound (1.3) and Kingman's ergodic theorem [22], it can be shown that the almost-sure limit exists and is equal to sup N ≥1 1 N E log Z (1,1)→(N,cN ) (β) . Though there are useful bounds for f c (β) (see for instance Comets [10]), explicit expressions for the its value remaining unknown but for a few cases which we now discuss.
First we consider the infinite temperature limit case β = 0. In this case the partition function Z (1,1)→(n,m) (0) is deterministic for every (n, m), simply counting the number of paths starting at (1, 1) and ending at (n, m). A straightforward computation using Stirling's formula tells us that the free energy f c (0) is given by We are also able to make sense of the zero temperature limit-the asymptotic case where β → ∞. Here, the partition function concentrates on the path maximising the energy F (π). Namely, for any pair of points x ≤ y in Z 2 , we have In particular, without too much concern at this stage for the technical details surrounding the interchange of limits, we have The value of g c is known in a few special cases. For example, in the case that each ω(z) is exponentially distributed with mean 1 Rost [37] showed that See [29] for further discussion of the single-path zero-temperature limit ℓ c .
Finally, there is one particular distribution for the random environment for which an explicit expression is known for f c (β)-at a positive and finite value of β. Namely, in the case where each e βω(z) has the inversegamma distribution with parameter µ as in (1.7), Seppäläinen [38] discovered a remarkable underlying algebraic structure based around the beta-gamma algebra making the partition function exactly solvable.

Asymptotics for finitely many long paths
For nice k-points x and y, we study the asymptotics of the partition function Z x→y+(N,cN ) (β) associated with the k-points stretched far apart in the (1, c) asymptotic choice of direction. Here we are able to exploit the subadditivity due to the series bound (1.3) to prove the following results.
Theorem 2.1. Let c be a positive rational. Then there exists a function f c (k, β) such that for any pair of nice k-points x and y, we have the almost sure convergence The limit satisfies (N,cN ) , and this quantity is independent of the choices x and y.
We point out that by (2.5) and Theorem 2.1, by definition we have f c (1, β) = f c (β). We also remark that by the parallel bound (1.5), it can be seen that the limits f c (k, β) satisfy the inequalities for positive integers k and j. We can say something much stronger however. According to the following theorem, f c (k, β) grows linearly in k.
Theorem 2.2. Let c be a positive rational. Then the limiting asymptotics of k-paths in a directed polymer satisfy Theorems 2.1 and 2.2 are proved in Section 5. We remark that the assumption that c is rational is a matter of convenience, and it is possible to prove that Theorems 2.1 and 2.2 holds for all positive c if we assume that the environment satisfies P (ω(0, 0) > L) = 1 for some real number L. We refer the reader to [34] and [35] for methods extending results from rational to irrational c in a related model.
In the next section we look at the case where there are many paths of finite length.

Asymptotics for many paths of finite length
Where in the last section we had finitely many paths and let their lengths tend to infinity, in this section we do the opposite, considering many non-intersecting paths of fixed length running side-by-side, and letting the number of paths tend to infinity. In this direction, for x ∈ Z 2 and h = (h 1 , h 2 ) ∈ Z 2 , define the stacked k-point x h↑k at x in the h direction by For x ≤ y in Z 2 , and a direction h such that h 1 ≤ 0 < h 2 , we now consider the the large k-asymptotics of the random variable Z x h↑k →y h↑k (β). The parallel bound (1.5) gives us a subadditivity in the k-variable which we use to prove the following result.
Then as k → ∞, the random variable 1 k log Z x h↑k →y h↑k (β) converges almost surely to a deterministic limit I y−x,h (β). In fact, in the infinite temperature case (β = 0), for certain choices of h, the asymptotic limit I z,h (0) can be computed explicitly by using the asymptotic theory of Toeplitz determinants [5]. In this direction we define the symbol associated with z and h to be the Laurent series a z,h : T → C on the unit circle given by Every continuous function a : T → C has a unique decomposition a(e it ) = |a(e it )|e ic(t) where c : [0, 2π) → R is a continuous function satisfying c(0) = 0. The winding number of a is the integer wind(a) := 1 2π lim t→2π c(t). (2.10) Using Szegö's limit theorem for the asymptotics of Toeplitz determinants, in Section 5.4 we prove the following result, which gives an expression for I z,h (0) in terms of the symbol a z,h .
Theorem 2.4. Suppose that h 1 < 0 < h 2 and the winding number of the symbol a y−x,h is zero. Then there is a Laurent series m∈Z c m s m satisfying In particular, I x→y,h (0) = c 0 .
We now obtain the exact asymptotics using Theorem 2.4. First we note that the symbol associated with x, y and h is given by We remark that Re a y−x,h (e it ) = 6 cos(t) + 10 for each t in [0, 2π). It follows that a y−x,h has positive real part on T, and hence has winding number 0. It is straightforward to show that the symbol has representation It then follows from Theorem 2.4 that I (3,2),(−2,2) = log(5 + 2 √ 5) ≈ log 9.472 ≤ log 10.

Asympotics for many long paths
Finally, in this section we consider the asymptotics of the partition functions associated with many paths in a rectangle whose dimensions are growing to infinity together with the number of paths. For k ≥ 1 and x ∈ Z 2 , we define the stacked k-point x ↑k above and below a point x by For integers m, n, k, consider k-points (1, 1) ↑k and (n, m) ↓k at the bottom-left and top-right corner of an m × n rectangle. Then there is at least one k-path from (1, 1) ↑k to (n, m) ↓k if and only if k ≤ m ∧ n.
With this picture in mind, we consider the scaling regime m = cN, n = N, k = αN for 0 < α ≤ c ≤ 1, leading us to study the asymptotic growth of the random variable To understand the 1 N 2 scaling, it is useful first to consider the infinite-temperature limit β = 0, which amounts to a computation calculating the asymptotic size of the set Γ (1,1) ↑k →(n,m) ↓k of k-paths on the m × n rectangle. Indeed, in Section 5.6, we exploit determinantal identities to prove the following result.
and Q(u) = u 2 2 log u. With the scaled-k high temperature limit at hand, in Section 5.5 we use Jensen's inquality to prove the following result relating the positive temperature partition function to the infinite-temperature limit.
Lemma 2.6. Let x and y be k-points such that x ≤ y, and let p(x,

Now define the upper and lower limits
The following result is an immediately corollary of Lemma 2.5 and Lemma 2.6, showing that these lower and upper limits may be sandwiched within terms involving the infinite temperature limit.
Corollary 2.7. The asymptotic k-path partition functions satisfy However, we conjecture the following stronger result.
Conjecture 2.8. The lower and upper limits R − c (α, β) and R + c (α, β) agree, and moreover in this case

Stochastic interfaces 3.1 Stochastic interfaces
Let Λ be a finite subset of Z 2 and let Λ * := x, y ∈ Λ 2 : y − x is equal to e 1 or e 2 be the set of directed edges in Λ. Suppose now V is a convex function on R and (W x : x ∈ Λ) are weight functions. A stochastic interface is a random function φ : Λ → R whose law is given by is the interface partition function. We observe that if D is a subset of Λ, then the marginal law of φ on D may be obtained by integrating out φ on Λ − D. That is, this marginal law is given by We will be particularly interested in stochastic interfaces on triangular and square subsets of Z 2 . Considering triangular sets first, let be the set of diagonal entries and let E N := T N − D N be the non-diagonal entries. When Π = D N , we call the energy integral a pattern integral and write g := g DN for short. In other words, a pattern integral is simply a function g V : R N → R given by where dφ(x) is Lebesgue measure and δ u is the Dirac mass at u.
Though the majority of surrounding literature on stochastic interfaces is restricted to models with symmetric interaction potentials, we will be most interested in the potentials associated with the exponential and bead interaction models, neither of which are symmetric. These are given by respectively, and will appear in stochastic interfaces relating to random polymers and random matrices which we introduce in the following two sections.
For a moment let the weight functions (W x : x ∈ T N ) be zero and consider interfaces defined on the triangle T N with the bead interaction V (u) = bead(u). It is straightforward to see that the functions φ : T N → R for which the Hamiltonian H TN [φ] is finite are precisely the Gelfand-Tsetlin patterns, namely those functions satisfying the inequalities Given a vector (λ 1 , . . . , λ N ), we write GT N (λ) for the set of Gelfand-Tsetlin patterns φ : T N → R satisfying (φ(1, 1), . . . , φ(N, N )) = (λ 1 , . . . , λ N ). Clearly this set is empty unless λ 1 ≥ . . . ≥ λ N . In fact, it is well-known that the pattern integral g bead associated with the V (u) = bead(u) is given by it is straightforward to prove the volume formula (3.3) by induction.
As for the exponential interaction potential exp, the associated pattern integral is known as a Whittaker function (with parameter 0).
We record the following lemma, which we will use shortly to connect the diagonal entries of stochastic interfaces with both the eigenvalue ensembles of random matrix theory as well as the so-called Whittaker measure. N )) is given by Proof. Integrating out the off-diagonal variables (φ(i, j)) 1≤i<j≤N and (φ(i, j)) 1≤j<i≤N and using the definition (3.1), we obtain two powers of g V (λ).
In the next section we discuss the Whittaker measure, a collection of random variables related to the partition functions of a random polymer which may be thought of in terms of a stochastic interface.

The Whittaker measure as a stochastic interface
We will see now that for a random polymer with product weights distributed according to the inverse gamma law (1.7), the partition functions have an expression in terms of a stochastic interface. In this direction, first recall the inverse gamma distribution with parameter µ and let F µ (s) := s 0 I µ (du) be the associated distribution function. In order to consider the simultaneous behaviour of our partition functions as µ varies, we would like to take a coupling of our weight variables as functions of µ. To this end, suppose U (z) : z ∈ Z 2 are uniformly distributed random variables on the unit interval, and for z ∈ Z 2 define the random variable ζ µ (z) := F −1 µ (U (z)). Clearly ζ µ (z) is inverse gamma distributed with parameter µ.
and consider the associated partition functions We emphasise that in contrast to the partition functions studied in Section 2, the initial points {(1, i − 1) : i = 1, . . . , k} are included in the weight products in (3.7). Now consider the random function ϕ N µ : with the conventions that τ N µ (m, 0) =τ N µ (m, 0) = 1 for each m. By way of a diagram the reader may convince themselves that τ N µ (N, k) =τ N µ (N, k) for every k ≤ N , from which it follows that the overlapping definitions in (3.8) on the diagonal i = j agree. We remark that (3.8) may be inverted to give It follows from results in [11] that ϕ N µ is a stochastic interface: The interface partition function is given by For completeness, we have included a proof of Proposition 3.2 in Section 6.1. We remark that using the definition g exp in Lemma 3.1, it follows from Theorem 3.2 that the marginal law of the diagonal (λ 1 , . . . , λ N ) := ϕ N µ (1, 1), . . . , ϕ N µ (N, N ) may be given in terms of the so-called Whittaker measure with constant parameter µ: (3.10)

Eigenvalue processes and µ → 0
where H is Hermitian and U is unitary. We define the eigenvalue process ϕ H,U : S N → R associated with H and U to be the function ϕ H,U : S N → R given by Let us remark that the N ×N matrices H and U * HU have the same eigenvalues, hence the two definitions in (3.11) are consistent on the diagonal i = j. We also point out that thanks to Cauchy's interlacing theorem, the eigenvalues of each A (k) interlace those of A (k+1) . This tells us that for any directed edge We now consider the eigenvalue processes associated with certain ensembles of unitarily invariant random matrices, introducing the Gaussian Unitary Ensemble (GUE) and the Laguerre Unitary Ensemble (LUE). We say a random N × N complex Hermitian matrix is GUE distributed if its law is given by and we say a random M × M complex Hermitian matrix is LUE distributed with underlying parameter N if its law is given by We will also use the following property of LUE matrices, which is straightforward to prove using the representation of Laguerre matrices as a product of matrices with independent complex Gaussian entries [14,Chapter 3].
Remark. If H is distributed with law P N LUE,N , then the marginal law of the minor H (k) is given by P N LUE,k . The following proposition, which states that the eigenvalue processes associated with GUE and LUE random matrices can be reformulated as stochastic interfaces, is a consequence of well known properties of the eigenvalues of unitarily invariant random matrices [3,20]. Then the eigenvalue process ϕ H,U associated with (H, U ) is a stochastic interface with interaction potential bead(u) and weight functions Alternatively, suppose H is LUE distributed and U is Haar distributed on the set of unitary matrices. Then the eigenvalue process ϕ H,U associated with (H, U ) is a stochastic interface with interaction potential bead(u) and weight functions We provide a proof of Proposition 3.3 in Section 6.1. For short, we refer to the eigenvalue processes occuring in Proposition 3.3 as the GUE and LUE eigenvalue processes. With this result at hand (in fact for the moment we only require the part regarding LUE matrices), we are ready to establish the relationship between the partition functions of the polymer with inverse-gamma distributed weights and the eigenvalues of random matrices.
Theorem 3.4. Let ϕ N µ : S N → R be the interface defined in (3.8). Then as µ ↓ 0, the rescaled process µϕ N µ converges in distribution to the LUE eigenvalue process. Proof. By Theorem 3.2, ϕ N µ is a stochastic interface with the exponential interaction potential and weight It is immediate that the change of variable µϕ N µ is also a stochastic interface with interaction potential V (u) = exp(u/µ) and weight function Taking µ ↓ 0, we have the (Lebesgue-almost-everywhere) convergence of both the interaction potential exp(u/µ) to bead(u) and the weight e −u/µ µ to ∞½ u<0 , which are precisely the interactions and weights associated with ϕ N LUE . We now consider the implications of this connection by looking directly at the small-µ asymptotics of the polymer. Suppose ζ µ is inverse-gamma distributed with parameter µ, and let χ µ := µ log ζ µ . Using the small-µ asymptotics of the Gamma function, it is straightforward to verify that That is, χ µ converges in distribution to a standard exponential random variable as µ ↓ 0. Applying these facts to the partition functions (3.7), we see that as µ ↓ 0 we have the convergence in distribution where (e(z) : z ∈ Z 2 ) are independent standard exponential random variables.
On the other hand, as µ ↓ 0, combining (3.9) and Theorem 3.4, we have the convergence in distribution Comparing (3.14) with (3.15) we obtain the distributional equality

The µ → ∞ limit
Where in the last section we showed that the small-µ asymptotics of the interface ϕ N µ , in this section we consider the large-µ asymptotics. The following theorem states that as µ → ∞, a suitable rescaling of the ϕ N µ converges to a explicit deterministic shape related to a combinatorial problem.
Theorem 3.6. Let θ N µ : S N → R be the rescaled interface Then as µ → ∞, θ N µ converges almost-surely to a deterministic function θ N min : S N → R, where θ min is the symmetric function given by Moreover, the function θ N min is the minimiser of the energy functional F : R SN → R given by Theorem 3.6 is proved in Section 6.2 by directly analysing the large-µ behaviour of the interface ϕ N µ , and studying the relationship this interface has with a deterministic polymer via the application of the law of large numbers to 1/ζ µ (z) (which for integer µ has a representation at the sum of µ independent exponential random variables). The fact that the function θ N min minimises F N [θ] is an offshoot of our result, and we suspect there is a more direct combinatorial proof.
That completes the section on finite interfaces. In the next section, we study the asymptotics of these interfaces as N → ∞.

Scaling limits of stochastic interfaces 4.1 Surface tension and asymptotics of stochastic interfaces
In this section we willl be interested in applying thermodynamic heuristics to study the macroscopic behaviour of the stochastic interfaces seen in the previous section, formulating their limiting shapes in terms of variational problems, and using these limit shapes to anticipate the asymptotics of the partition functions associated with the non-intersecting log-gamma polymer.
We begin by introducing surface tension, a concept first developed by Wulff in his study of crystal interfaces [43]. Let ∂S N and S • N denote the interior and the boundary of the square S N . Consider the square Hamiltonian associated with an interaction potential V and W x ≡ 0 for every x. For N ≥ 3, we define the finite surface tension of interaction potential V at tilt p := (p 1 , p 2 ) ∈ R 2 by (4.1) The following result by Funaki and Spohn [16] states that under relatively strong conditions on the interaction potential V , the finite surface tensions σ V N converge to a limit as a N → ∞. The limit σ V is called the surface tension associated with V .
Proposition 4.1. Suppose the potential function V is symmetric, twice differentiable, and satisfies c − ≤ V ′′ (x) ≤ c + for some positive reals c − ≤ c + . Then the limit σ V (p) := lim N →∞ σ V N (p) exists and is a convex function on R 2 .
Neither of the interaction potentials bead and exp are symmetric, nor do they satisfy the technical condition required by Funaki and Spohn for the existence of the associated surface tension. Nonetheless, it was established by Sun [41] that we have the existence of the surface tension associated with the bead(u) interaction, and moreover, Sun provided a formula for σ bead (p) in a different coordinate system which we discuss below. Based on the relative similarity between the exponential and bead interaction, we anticipate the following conjecture. With the definition of surface tension at hand, we are ready to study the macroscopics of stochastic interfaces. First we define a rescaling of stochastic interfaces from S N to the unit square in R 2 .
where s ′ is the smallest multiple of 1/N greater than s, and t ′ is defined similarly.
In Chapter 6 of [15], Funaki shows the asymptotics of stochastic interfaces with potentials satisfying the conditions in Proposition 4.1 may be given in terms of minimisers of variational problems. Furthermore, it is known that the asymptotic shapes of interfaces with the bead interaction potential have intimate relationships with models in free probability, see for instance Metcalfe [25]. With these observations in mind, and in the event that Conjecture 4.2 holds, we are lead to further predict the following result about the asymptotic shape of a certain class of interfaces.
where f N be a function satisfying f N (u) → ∞½ u / ∈A as N → ∞, and µ is a positive real number. Suppose further that if Z N is the partition function associated with the stochastic interface on S N , then the sequence Then the rescaled interface converges pointwise almost surely to a deterministic limit-that is, each (s, t) in S,φ N (s, t) converges almost surely to a deterministic limit ξ(s, t). Moreover, the limit function ξ is the minimiser in C 1 (S) of the functional E : C 1 (S) → R given by Finally, the value of the functional at the minimiser is given by E[ξ] = 0.

Asymptotics of the Whittaker measure
Now we consider the implications of Conjecture 4.4 for the function ϕ N µ defined in (3.8) and appearing as a stochastic interface in Proposition 3.2, with a particular focus on what this tells us about large-N asymptotics of the partition functions τ N µ (m, k) under the scaling m = cN , k = αN .
Indeed, we anticipate that the rescaled interfaceφ N µ converges pointwise almost-surely to a deterministic limit ξ µ , and that ξ µ is the minimiser over all functions v in C 1 (S) satisfying v(1, 1) ≥ 0 of the energy functional E µ : In this case, we are able to recover the large-N asymptotics of the many-path partition functions associated with the inverse gamma polymer of parameter µ. Namely, the large-N asymptotics of (3.9) suggest that Short of offering an explicit expression for the limit shape ξ µ , we make a few predictions about its properties. First of all, by the symmetry of the energy functional E µ (4.3), it is plain that the minimiser ξ µ is itself symmetric.
Moreover, consider minimising the two competing terms appearing in E µ [v] over functions satisfying v(1, 1) = 0. On the one hand, the first term S σ exp (∇v) encourages v to have negative derivatives with respect to both s and t, where as the intermediate term µ 1 0 v(s, s)ds wants v to decreases. As µ becomes larger, this second effect becomes stronger, and we anticipate that ξ µ is monotone decreasing in µ.
With these broad observations made, we now turn to discussing existing results in the literature which give us the values of ξ µ on the boundaries of the square S.
We now seek to understand the values taken by ξ µ on the other boundaries of the square S, namely points of the form (t, 1) and (1, t). To this end, again by (3.8) we have . [30]. In particular, it follows from Equation (2.1) and Proposition 2.1 of [30] that we have the relation
is simply a product of mN independent inverse-gamma random variables with parameter µ, and it follows from the law of large numbers that when m = cN , we have the almost sure convergence Now note that Combining (4.11) and (4.12) with (4.4), we yield (4.10).

The small-µ asymptotics of ξ µ and the Marčenko-Pastur law
We now consider the asymptotics of the limit shape ξ µ : S → R as µ → 0. Taking c = 1 − t + s and α = s, and using the definition of the eigenvalue process ϕ N LUE , we yield the following proposition . (4.14) On the other hand, assuming Conjecture 4.4, the representation of the eigenvalue process ϕ N LUE as an interface in (3.3) implies that the asymptotic limit ξ mp is the minimiser of the energy functional Reiterating the connection with the interface ϕ N µ , Theorem 4.5 can be read as saying where the convergence is pointwise almost sure on S. Assuming we can interchange the order of taking the limits µ → 0 and N → ∞ in the final term of (4.15), we anticipate that Finally, we now show that Proposition 4.5 implies Rost's equation (2.7) for the asymptotics associated with last passage percolation on an exponential polymer. Namely, taking µ ↓ 0 in (4.5), using (4.16) and (3.14) with k = 1, we obtain where ℓ c is defined as in (2.6). It remains to note from (4.14) and (4.13) that as required.

The large-µ asymptotics of ξ µ
We now consider the implications of our discussion in Section 3.4 as N → ∞. Let θ N min : S N → R be the function defined in (3.18), and letθ N min : S → R be the associated rescaling. By using Stirling's formula with the definition (3.18) of θ N min , it is possible to prove the following result. We provide details in Section 7.1.
Theorem 4.6. Let q(u) = u log u. The limit ξ ht (s, t) := lim N →∞θ N min (s, t) exists, and is given by the symmetric function satisfying Again, assuming we can interchange the limits N → ∞ and µ → ∞, by (3.17) we expect that (4.17) Finally, in light of the fact that θ N min minimises the energy functional (3.18), it is natural to expect the rescaled limit interface ξ ht : S → R is the minimiser of the energy functional where σ ht (∇v(s, t)) = exp ∂v ∂s + exp ∂v ∂t . We regard the energy functional E ht as an asymptotic analogue of the discrete functional in (3.19).
We take a moment to recapitulate on the large and small-µ limits of ξ µ we have seen, and on their related variational problems. The function ξ µ , defined as a limit of the rescaled interface ϕ N µ , is the minimiser of the energy functional S σ exp (∇v(s, t) Anticipating that taking µ to either 0 or ∞ commutes with taking N → ∞, we predicted the following. On the one hand, as µ → 0, we expect that µξ µ converges to ξ mp , which has an explicit expression (4.14) in terms of the Marcenko-Pastur distribution, and we predict is the minimiser of S σ bead (∇v(s, t))dsdt On the other hand, as µ → ∞, we expect that ξ µ (s, t) + (2 − s − t) log µ converges to ξ ht , which has the explicit expression (3.18) above, and we predict is the minimiser of S σ ht (∇v(s, t))dsdt +

Gaussian fluctuations at high temperature
In the last Section we looked at taking µ → ∞ after N → ∞, anticipating the convergence to a limit shape. In this section, we develop a finer picture, using the central limit theorem as a refinement of the law of large numbers.
Under a certain scaling limit, we are able to characterise the fluctuations of the partition functions τ N µ (m, k) at high temperature. Let Q N m,k be the uniform law on Γ N (m, k), and let π be a random variable with law Q N m,k . We make several assumptions about the asymptotic density of scaled k-paths in the N × N square. First, for (u, v) ∈ S and α ≤ c ≤ 1, we expect the existence of the limit αN ((uN, vN ) ∈ π) =: q c,α (u, v). Moreover, for any n distinct points (u 1 , v 1 ), . . . , (u n , v n ) in S we anticipate that we have the asymptotic decoupling This decoupling equation (4.19) can be seen to hold, at least in a weak sense, via the well-known bijection between non-intersecting paths and lozenge tilings of a hexagon, and in fact one can write down an explicit formula for the limiting function q c,α (u, v) via a formula given for the limit shape of the corresponding tiling model in [9]. We refer the reader also to the work of Johansson [19], where the non-intersecting paths model is related to an extended Hahn process.
Let κ > 0 be a constant, and consider the sequence of random processes , k), where m = ⌊tN ⌋ and k = ⌊sN ⌋. In Section 7.2, we sketch ideas leading us to the following conjecture.
Conjecture 4.7. The limit (4.18) exists and the satifies the asymptotic decoupling (4.19). Furthermore, the random process {H N κ (s, t) : 0 ≤ s ≤ t ≤ 1} converges in distribution to a centred Gaussian process {H κ (s, t)} with covariance where f, g L 2 (S) := S f (u, v)g(u, v)dudv is the L 2 inner product on the unit square S.
It is immediate from (4.18) that q c,c (u, v) = ½ v<c in L 2 (S), and in particular, for any t,t in [0, 1] we have q t,t , qt ,t = t ∧t. This observation implies that

The semicircle law and surface tension in the bead model
In Section 8 we combine tools from random matrix theory with our variational approach to macroscopics of stochastic interfaces to obtain the following explicit expression for the surface tension associated with bead model where σ bead tilted is the surface tension in the change of coordinates given by σ V tilted (p, q) := σ V 1 2 p − q, 1 2 p + q .
An equivalent formula to (4.21) was obtained in Sun [41], where it is proved by viewing the bead model as a continuous version of the Cohn-Kenyon-Propp [8] dimer model. Sun uses Kasteleyn theory [21] to express the partition functions of this dimer model in terms of a contour integral, and studies the asymptotics of these contour integrals to obtain the expression [41,Definition 5.4].
Our approach, which we now overview, uses simpler technology from variational analysis and random matrix theory. On the one hand, we have the following result concerning the asymptotics of ϕ N GUE , which is an immediate consequence of the semicircle law of classical random matrix theory [1]. With these observations at hand, the main idea of our derivation is straightforward: if the minimiser of (4.23) has the form (4.22), then the tilted surface tension of the bead model must be given by (4.21).
To sketch out the key steps here, we develop the following scaling limit of the formula Combining (4.27) with (4.28), we obtain the following thermodynamic analogue of the Gelfand-Tsetlin volume formula However, we also know that v * must also be related to the semicircle law as it appears in (4.22). By plugging ξ sc into the differential equation (4.30) , and using a homogoneity property of the surface tension, we determine σ bead tilted as having the form in (4.21). The full details of this argument are given in Section 8, though it bares remarking here that in pinciple we could have alternatively used the Marčenko-Pastur distribution in place of the semi-circle law, and we only opt to use the latter because the calculations involved are more straightforward.

The Finite k case
This section is dedicated to proving the results stated in Section 2. We begin by proving Theorem 2.1. In Sections 5.1, 5.2 and 5.3 we will write Z(x → y) for Z x→y (β), since the k-poiints x and y appearing in these sections can be notationally heavy, and we won't be considering different values of β.

Proof of Theorem 2.1
The proof of Theorem 2.1 is based on a subadditivity argument. To sketch the main idea here, by the series bound (1.3), the random variables Y x→y := − log Z(x → y) are subadditive with respect to the ordering ≤ of nice points Z 2×k in the sense that for any three nice k-points x ≤ y ≤ z, we have We want to exploit this subadditivity to study the asymptotics of the random variable Z x→y+ (N,cN ) as N tends to infinity. To do so, we recall Kingman's subadditive ergodic theorem, giving a slight restatement of the theorem as it appears in Kingman's original paper [22].
Theorem 5.1. Let (X r,s ) r<s∈Z be a family of random variables defined on a probability space satisfying 1. For every triple of integers r < s < t, we have X r,t ≤ X r,s + X s,t .
2. For every a ∈ Z, the joint distribution of the processes (X r+a,s+a ) 0≤r<s are the same as those of (X r,s ) 0≤r<s .
3. The expectation g r := E[X 0,r ] exists, and there exists a real number L such that gr r ≥ Lr for every r ≥ 1.
Then for any r ∈ Z, the limit θ := lim s→∞ 1 s X r,s exists with probability one and in expectation, and furthermore θ = inf r≥1 1 r g r . With Kingman's subadditive ergodic theorem now stated, the proof of Theorem 2.1 is split into two parts. First we show that if p and q are positive integers such that q/p = c, then a result almost identical to Theorem 2.1 holds for all k-points x and y of the form (ap, aq) ↑k for some integer k. We then lift this restriction, proving Theorem 2.1 for all nice k-points.
Lemma 5.2. Let p and q be positive integers no smaller than k such that q/p = c. For integers r < s, define the random variables Then there is a real number f c (k, β) such that for any r, the random variables 1 s X r,s converge almostsurely to −f c (k, β). Moreover f c (k, β) is independent of the choice p and q, and f c (k, β) is equal to Proof. First we show that with X r,s defined as in (5.2), we are in the set up of Theorem 5.1. First of all, the inequality X r,t ≤ X r,s + X s,t is a consequence of (5.1). The second condition is immediate from the definition of X r,s (using the fact that the polymer weights are independent and identically distributed). Finally, we can show the final condition is satisfied using Jensen's inequality as follows. We have where Q (0,0) ↑k →(rp,rq) ↑k is the uniform measure on Γ (0,0) ↑k →(rp,rq) ↑k , andπ has law Q (0,0) ↑k →(rp,rq) ↑k . Using Jensen's inequality to interchange the order of E and log, we obtain Finally, using the inequality n k k ≤ n k ≤ en k k , it is straightforward to see that 1 r log #Γ (0,0→(rp,rq) is bounded, and hence there is a real L such that E[X 0,r ] ≥ Lr. This establishes that all three conditions in Theorem 5.1 hold for the doubly indexed sequence (X r,s ) r<s∈Z .
It follows that for each r ∈ Z, the limit It remains to prove f p,q (k, β) is independent of the choice of p and q. To see this, first note that we may assume without loss of generality that r = 0 since f p,q (k, β) is independent of r. In this case the random variables (W p,q s ) s≥1 given by converge almost surely to f p,q (k, β) as s → ∞. Now suppose p ′ and q ′ are another pair of integers no smaller than k such that q ′ /p ′ = c. Then there exist positive integers n, n ′ such that np = n ′ p ′ , and it follows that W p,q ns = W p ′ ,q ′ n ′ s for every s ≥ 0. It follows that we have where the limits in (5.3) are almost-sure limits. This ensures that f p,q (k, β) is independent of the choice of p and q, and justifies us hereafter writing f c (k, β) for this quantity.
We are now ready to prove Theorem 2.1. The main idea of the proof is that any nice k-points x and y + (N, cN ) may each be sandwiched between two points of the form (ap, aq), where a is some integer.
The following two simple lemmas are integral to the proof.
Lemma 5.3. For any collection of nice k-points x − ≤ x ≤ x + ≤ y − ≤ y ≤ y + , we have the sandwich bound for Z(x → y): .
Proof. The upper bound is derived by rearranging the series bound for the points x − ≤ x ≤ y ≤ y + , where as the lower bound is immediate from the series bound for the points x ≤ x + ≤ y − ≤ y.
The reader will immediately convince themselves that the following lemma is true by means of a picture. We are now ready to prove Theorem 2.1, which we recall states that for any pair of nice points x and y, 1 N log Z (x → y + (N, cN )) converges almost surely to f c (k, β) .
Proof of Theorem 2.1. Let x and y be nice k-points, and let p and q be a pair of positive integers not smaller than k such that q/p = c. By Lemma 5.4, there exist integers a < a ′ x − := (ap, aq) ↑k ≤ x ≤ (a ′ p, a ′ q) ↑k =: x + .
In particular, using (5.5) and (5.6) in the sandwich bound (5.4), we have where It is a straightforward consequence of Lemma 5.2 that both 1 αN p m N and 1 αN p M N converge almost surely to f c (k, β) as N → ∞. The conclusions of Theorem 2.1 now follow from (5.7).

Proof of Theorem 2.2
We now prove Theorem 2.2, which states that f c (k, β) = kf c (β). The proof is split into two inequalities, the first of which is significantly easier than the other.
To prove the reverse inequality f c (k, β) ≥ kf c (β), first we require a quick lemma giving a blunt lower bound on E[log Z x→y ] in terms of the distance between k-points x and y. Lemma 5.5. Let x ≤ y be k-points. Then is the expectation of the environment. Proof. Since x ≤ y, there is a k-path π from x to y, and in particular Since π contains p(x, y) identically distributed weights, completing the proof.
Proof of the inequality f c (k, β) ≥ kf c (β). Let x be the stacked point x = (−k − 1, 0) ↑k , and let y be the stacked point (k + 1, 0) ↑k . Write c = q/p as a quotient of positive integers, and for integers M consider the partition function Z x → y + (M 2 p, M 2 q) . On the one hand, by Theorem 2.1 we know that On the other hand, consider the sequence of k-points where each k-points z l in the sequence is given by First we consider A M . Note that by Lemma (5.5) It remains to study the asymptotics of B M . First, we note by construction that any k-tuple of paths from z i toẑ i+1 are necessarily non-intersecting, from which it follows that Moreover every l and i the random variables Z z l i →ẑ l+1 i are identically distributed with the same law as Z ((0, 0) → (M p, M q − 1)). In particular, Let ǫ > 0. By applying Theorem 2.1 with k = 1, x = (0, 0) and y = (0, −1), there exists M 0 such that In particular, for all M ≥ M 0 , Since ǫ and M are arbitrary, we yield , proving the result.

Proof of Theorem 2.3
We now give prove Theorem 2.3, the proof of which is straightforward.
Provided E[e βω1,1 ] < ∞, we are in the set up of Kingman's subaditive ergodic theorem, Theorem 5.1, and hence 1 k X 0,k = 1 k log Z x h↑k →y h↑k converges to a deterministic limit J x,y,h (β) depending on x, y, h, and β. Since the weights of the polymer are independent, it is plain that J only depends on x and y through the difference y − x. We call the limit I y−x,h (β). This completes the proof of Theorem 2.3.
In the sequel we return to writing Z x→y (β) (as opposed to Z(x → y)) for the x to y partition function.

Proof of Theorem 2.4
To prove Theorem 2.4, we recall Szegö's limit theorem, following Böttcher and Silbermann [5,Chapter 5] and Bump [6,Chapter 42]. (Z xi→yj (0)). (5.10) In order to exploit the successful theory of the asymptotics of Toeplitz determinants in order to study the large-k asymptotics of polymer partition functions, we are interested in the k-points x and y such that Z xi→yj (0) is a function of j − i only.
In order for the right-hand-side of (5.10) to be a Toeplitz matrix, there must be a pair of base points x and y in Z 2 as well as an h ∈ Z 2 such that When x and y take the form (5.11), it is clear that where for m ∈ Z, In particular, Whenever

The infinite temperature sandwich bounds
In this section we prove Lemma 2.6 Proof of Lemma 2.6. We remark that the infinite temperature partition function Z x→y (0) counts the set Γ x→y of k-paths from x → y. Let Q x→y be the uniform measure on Γ x→y , and let Π be a k-path-valued random variable with law Q x→y . Clearly, x∈πi−{xi} ω(x). In particular, We remark that under the measure PQ x→y , F (Π) is equal in law to the sum of p(x, y) independent and identically distributed random variables with law ω(0, 0). Now on the one hand, using Jensen's inequality to interchange the order of P and ln, we obtain where G(β) = E[e βω(0,0) ]. This yields the upper bound. On the other hand, using Jensen's inequality to interchange the order of ln and Q x→y we have 1 p(x, y) P ln Q x→y e βF (Π) ≥ 1 p(x, y) PQ x→y ln e βF (Π) giving the lower bound.

The asymptotics of the high-temperature scaled-k limit
This section is devoted to proving Lemma 2.5. In this direction, first we stateand, for completeness, include a proof ofthe following result regarding the infinite temperature partition function on a finite rectangle. We note that, via the well-known bijection between non-intersecting paths and plane partitions, this is equivalent to Macmahons formula. We refer the reader to Stanley [40,Section 7.20] for an approach using the RSK correspondence.
Proposition 5.7. Let m, n, k be positive integers satisfying k ≤ m ∧ n. Then where H(N ) := N −1 j=0 j! is the superfactorial. Proof. Let m, n, k be positive intgers satisfying k ≤ m ∧ n, and suppose π = (π 1 , . . . , π k ) is a k-path going from (1, 1) ↑k → (n, m) ↓k . It is easily seen that for each i, the i th path π i must go through the points x i = (k + 1 − i, i) and y i := (m − i + 1, n − k + i). Let x = (x 1 , . . . , x k ) and y = (y 1 , . . . , y k ) be the associated k-points. Then we have the following identity for the rectangular zero-temperature partition functon Z (1,1) ↑k →(n,m) ↓k (0) = Z x→y (0). (5.13) By the Lindström-Gessel-Viennot formula (2.4), Now note that the partition function Z xi→yj (0) is simply counts the number of paths from x i to y j , that is 14) It bares remarking at this stage that the right-hand-side of (5.13) can be expanded as a more tractable determinant than the left-hand-side, since the path length from each x i to each y i is m + n − 2k, which doesn't depend on i or j.
Proof of Lemma 2.5. To prove Lemma 2.5, it remains to study the large-N asymptotics of (5.12) when m = cN , n = N and k = αN , for any c > 0 and α ≤ c ∧ 1. By adapting (4.26), we see that for p > 0, Now the result follows from (5.12) and (5.17), noting the identity 6 Proofs of results in Section 3

Proofs of Proposition 3.2 and Proposition 3.3
In this section we prove Propositions 3.2 and 3.3, which state that certain random functions related to random polymers and random matrices may be expressed as stochastic interfaces.
which by Lemma 3.1, agrees with the law of the corresponding stochastic interface model. It remains to show that the processes have the same law not just on the diagonal but everywhere on S N . In this direction, by [11,Theorem 3.7 (ii)], the conditional law of {ϕ N µ (i.j) : 1 ≤ i ≤ j ≤ N } given the values on the diagonal ϕ N µ (1, 1), . . . , ϕ N µ (N, N ) = (λ 1 , . . . , λ N ) is given by Finally, it follows from the construction in [11] that the random variables {τ N (m, k)} 1≤k≤m≤N are conditionally independent of the random variables {τ N (m, k)} 1≤k≤m≤N given the diagonal. By the distributional symmetry of ϕ N µ on either side of the diagonal, this completes the proof.
Equation (6.1) is known as the Ginibre formula, and it is well known that the N eigenvalues eigenvalues λ 1 > . . . > λ N of H of a matrix from the Gaussian Unitary Ensemble also have this law, establishing that the law of the stochastic interface ϕ N agrees with the eigenvalue process ϕ N H,U on the diagonal i = j.
To prove that the equality in distribution for ϕ N H,U and ϕ N holds not just on the diagonal but everywhere on the square S N , we appeal to the results of Baryshnikov [3], which state that if a random matrix H is invariant under unitary conjugations, then the eigenvalue process is a 'uniform lift' of the diagonal (λ 1 , . . . , λ N ) onto the Gelfand-Tsetlin pattern GT N (λ).

Proof of Theorem 3.6
In this section we consider the large-µ asymptotics of the interface ϕ N µ , providing a proof of Theorem 3.6.
Proof of Theorem 3.6. By definition, the law of the interface ϕ N µ : We want to take a change of variables so that the interaction term competes with the weight term for large µ. Consider the change of variables It is straightforward to show that the random function θ N µ : S N → R is itself a stochastic interface whose law is proportional to exp(−F N [θ]), where Plainly, as µ to ∞ the interface θ N µ converges in distribution to the deterministic function θ N min : That completes the proof that θ N µ converges in distribution to a deterministic function θ N min minimising a variational problem. It remains to show that this minimiser has the form (3.18).
To do this, we consider the direct implications for the polymer partition functions of sending µ → ∞. For integer values of µ, 1 ζµ(z) can be written as a sum of µ independent and identically distributed exponential variables with mean 1. It follows that as µ → ∞ 1 µζ µ (z) converges in distribution to 1. (6.4) Since each product in the sum τ N µ (m, k) := π∈Γ N (m,k) z∈π ζ µ (z) contains k(N + m − k) weights, we have Now using the definitions of ϕ N µ and θ N µ (which appear in (3.8) and (6.2) respectively), for i ≤ j we have .
It follows that Combining (6.8) with (6.9), and using the definition H(n) := N i=1 (i − 1)! of the superfactorial, we obtain the formula Finally, it is clear from the distributional symmetry of ξ N µ , θ N min (i, j) = θ N min (j, i).
7 Proofs of results in Section 4 7.1 Proof of Theorem 4.6 In this section we prove Theorem 4.6.
Proof of Theorem 4.6. Since θ N min is symmetric, any potential limit function is also symmetric, so without loss of generality we consider {s ≤ t}. In this case, by (3.18) and Definition 4.3, the rescaled interface associated with θ N min is given bȳ By Stirling's formula, 1 N log(⌊uN ⌋) = u(log N − 1) + q(u) + o(1), where q(u) := u log u. Using the fact that it follows thatθ N min (s, t) → ξ ht (s, t), given by

Calculations surrounding Conjecture 4.7
Section 7.2 is dedicated to sketching calculations leading us to Conjecture 4.7, which anticipaties Gaussian fluctuations of the logarithmic partition functions of the log-gamma polymer at high temperature.
First of all, consider that by the central limit theorem, for each z, as µ → ∞ the random variable converges in distribution to a standard normal random variable. Now define the variable r z (µ) through the equation It is straightforward to check that r z (µ) = 1 + µ −1/2 N µ (z) −1 N z (µ), and hence r µ (z) also converges in distribution to a standard normal random variable as µ → ∞.
Assuming (4.18) and (4.19), we now consider the large-µ-large-N asymptotics of the partiton functions under a suitable scaling. Consider the random variable Using (7.1) and the fact that each path in Γ N (m, k) contains k(N + m − k) weights, we have where the sum in the final line above is taken over all subsets F of the square S N := {1, . . . , N } 2 . Expanding further, and considering the n! ways of ordering the elements of a set F of size n, we have where the internal sum is taken over all distinct n-tuples of elements of S N . When N is large, for each n the set of non-distinct k-tuples S n N − S Taking the scaling µ = κN 2 , and using (7.3), we anticipate that whereẆ is a space-time white noise on S independent of c and α. We remark that by consideringτ in place of τ , we also expect logG N κN 2 (cN, αN ) := log µ k(N +m−k)τ N µ (m, k) #Γ N (m, k) converges in distribution to 1 κ Sw c,α (u, v)Ẇ (u, v), wherew is defined in analogy to (4.18), instead using the uniform measureQ N m,k onΓ N (m, k).
Let A : C 1 ([0, 1]) → R be a functional. We define the functional derivative D A of A to be the map D A : C 1 ([0, 1]) × C 1 ([0, 1]) → R given by It is easily verified that D A is linear in the second argument for every ρ, and hence by the Riesz representation theorem [36], for every ρ there exists a Radon measure Λ A (ρ, ·) on [0, 1] such that D A (ρ, η) = 1 0 η(r)Λ A (ρ, dr).
We call the measure Λ A (ρ, ·) the Riesz measure associated with A and ρ. Note that whenever ρ is a local minima or maxima of the functional A, the Riesz measure Λ A (ρ, ·) is zero.
Finally, we remark that whenever A : C 1 ([0, 1]) → R is itself a linear map, then We will be interested in studying the functional derivatives of minimal Wulff functionals. Recall the tilted change of coordinates (8.1). Since this change of coordinates has unit Jacobian, we may rewrite the both the Wulff functional as a functional on C 1 (U ) by It it straightforward to verify (using (8.6)) that D G (f, η) = 1 0 η(r)σ tilted (ρ ′ (r), f (r))dr + D M (ρ, η).
By a standard variational argument letting η(r)dr approximate a dirac mass, the equation (8.8) follows.
It follows from (8.8) and the differentiated homogeneity property (8.17) that the surface tension σ bead tilted , v * , and the Riesz density (8.7) must jointly satisfy