Passage Percolation Research Articles

Transition between characters of classical groups, decomposition of Gelfand-Tsetlin patterns and last passage percolation

We study the combinatorial structure of the irreducible characters of the classical groups GLn(C), SO2n+1(C), Sp2n(C), SO2n(C) and the “non-classical” odd symplectic group Sp2n+1(C), finding new connections to the probabilistic model of Last Passage Percolation (LPP). Perturbing the expressions of these characters as generating functions of Gelfand-Tsetlin patterns, we produce two families of symmetric polynomials that interpolate between characters of Sp2n(C) and SO2n+1(C) and between characters of SO2n(C) and SO2n+1(C). We identify the first family as a one-parameter specialization of Koornwinder polynomials, for which we thus provide a novel combinatorial structure; on the other hand, the second family appears to be new. We next develop a method of Gelfand-Tsetlin pattern decomposition to establish identities between all these polynomials that, in the case of irreducible characters, can be viewed as branching rules. Through these formulas we connect orthogonal and symplectic characters, and more generally the interpolating polynomials, to LPP models with various symmetries, thus going beyond the link with classical Schur polynomials originally found by Baik and Rains (Duke Math. J., 2001). Taking the scaling limit of the LPP models, we finally provide an explanation of why the Tracy-Widom GOE and GSE distributions from random matrix theory admit formulations in terms of both Fredholm determinants and Fredholm Pfaffians.

Brownian regularity for the Airy line ensemble, and multi-polymer watermelons in Brownian last passage percolation

The Airy line ensemble is a positive-integer indexed system of random continuous curves whose finite dimensional distributions are given by the multi-line Airy process. It is a natural object in the KPZ universality class: for example, its highest curve, the Airy 2 _2 process, describes after the subtraction of a parabola the limiting law of the scaled energy of a geodesic running from the origin to a variable point on an anti-diagonal line in such problems as Poissonian last passage percolation. The ensemble of curves resulting from the Airy line ensemble after the subtraction of the same parabola enjoys a simple and explicit spatial Markov property, the Brownian Gibbs property. In this paper, we employ the Brownian Gibbs property to make a close comparison between the Airy line ensemble’s curves after affine shift and Brownian bridge, proving the finiteness of a superpolynomially growing moment bound on Radon-Nikodym derivatives. We also determine the value of a natural exponent describing in Brownian last passage percolation the decay in probability for the existence of several near geodesics that are disjoint except for their common endpoints, where the notion of ‘near’ refers to a small deficit in scaled geodesic energy, with the parameter specifying this nearness tending to zero. To prove both results, we introduce a technique that may be useful elsewhere for finding upper bounds on probabilities of events concerning random systems of curves enjoying the Brownian Gibbs property. Several results in this article play a fundamental role in a further study of Brownian last passage percolation in three companion papers (Hammond 2017a,b,c), in which geodesic coalescence and geodesic energy profiles are investigated in scaled coordinates.

Interlacing and Scaling Exponents for the Geodesic Watermelon in Last Passage Percolation

In a discrete planar last passage percolation (LPP), random values are assigned independently to each vertex in $${\mathbb {Z}}^2$$ , and each finite upright path in $${\mathbb {Z}}^2$$ is ascribed the weight given by the sum of values attached to the vertices of the path. The weight of a collection of disjoint paths is the sum of its members’ weights. The notion of a geodesic, namely a path of maximum weight between two vertices, has a natural generalization concerning several disjoint paths. Indeed, a k-geodesic watermelon in $$[1,n]^2\cap {\mathbb {Z}}^2$$ is a collection of k disjoint upright paths contained in this square that has maximum weight among all such collections. While the weights of such collections are known to be important objects, the maximizing paths have remained largely unexplored beyond the $$k=1$$ case. For exactly solvable models, such as exponential and geometric LPP, it is well known that for $$k=1$$ the exponents that govern fluctuation in weight and transversal distance are 1/3 and 2/3; which is to say, the weight of the geodesic on the route $$(1,1) \rightarrow (n,n)$$ typically fluctuates around a dominant linear growth of the form $$\mu n$$ by the order of $$n^{1/3}$$ ; and the maximum Euclidean distance of the geodesic from the diagonal typically has order $$n^{2/3}$$ . Assuming a strong but local form of convexity and one-point moderate deviation estimates for the geodesic weight profile—which are available in all known exactly solvable models—we establish that, typically, the k-geodesic watermelon’s weight falls below $$\mu n k$$ by order $$k^{5/3}n^{1/3}$$ , and its transversal fluctuation is of order $$k^{1/3}n^{2/3}$$ . Our arguments crucially rely on, and develop, a remarkable deterministic interlacing property that the watermelons admit. Our methods also yield sharp rigidity estimates for naturally associated point processes. These bounds improve on estimates obtained by applying tools from the theory of determinantal point processes available in the integrable setting.

Absorbing time asymptotics in the oriented swap process

The oriented swap process is a natural directed random walk on the symmetric group that can be interpreted as a multispecies version of the totally asymmetric simple exclusion process (TASEP) on a finite interval. An open problem from a 2009 paper of Angel, Holroyd, and Romik asks for the limiting distribution of the absorbing time of the process as the number of particles goes to infinity. We resolve this question by proving that this random variable satisfies GOE Tracy–Widom asymptotics. As a central ingredient of our proof, we reexamine a distributional identity relating the behavior of the oriented swap process to last passage percolation, conjectured in a recent paper of Bisi, Cunden, Gibbons, and Romik. We use a shift-invariance principle for multispecies TASEPs, obtained by exploiting recent results of Borodin, Gorin, and Wheeler for the stochastic colored six-vertex model, to prove a weakened form of the Bisi et al. conjectural identity, that is nonetheless sufficient for proving the asymptotic result for the absorbing time.

One-point distribution of the geodesic in directed last passage percolation

We consider the geodesic of the directed last passage percolation with iid exponential weights. We find the explicit one-point distribution of the geodesic location joint with the last passage times, and its limit as the parameters go to infinity under the KPZ scaling.

The half-space Airy stat process

We study the multipoint distribution of stationary half-space last passage percolation with exponentially weighted times. We derive both finite-size and asymptotic results for this distribution. In the latter case we observe a new one-parameter process we call half-space Airy stat. It is a one-parameter generalization of the Airy stat process of Baik–Ferrari–Péché, which is recovered far away from the diagonal. All these results extend the one-point results previously proven by the authors.

Hidden invariance of last passage percolation and directed polymers

Last passage percolation and directed polymer models on ℤ 2 are invariant under translation and certain reflections. When these models have an integrable structure coming from either the RSK correspondence or the geometric RSK correspondence (e.g., geometric last passage percolation or the log-gamma polymer), we show that these basic invariances can be combined with a decoupling property to yield a rich new set of symmetries. Among other results, we prove shift and rearrangement invariance statements for last passage times, geodesic locations, disjointness probabilities, polymer partition functions and quenched polymer measures. We also use our framework to find “scrambled” versions of the classical RSK correspondence and to find an RSK correspondence for moon polyominoes. The results extend to limiting models, including the KPZ equation and the Airy sheet.

Fluctuations of transverse increments in two-dimensional first passage percolation

We consider a model of first passage percolation (FPP) where the nearest-neighbor edges of the standard two-dimensional Euclidean lattice are equipped with random variables. These variables are i.i.d. nonnegative, continuous, and have a finite moment generating function in a neighborhood of 0. We derive consequences about transverse increments of passage times, assuming the model satisfies certain properties. Approximately, the assumed properties are the following: We assume that the standard deviation of the passage time on scale r is of some order σ(r), and σ(r),r>0 grows approximately as a power of r. Also, the tails of the passage time distributions for distance r satisfy an exponential bound on a scale σ(r) uniformly over r. In addition, the boundary of the limit shape in a neighborhood of some fixed direction θ has a uniform quadratic curvature. By transverse increment we mean the difference between passage times from the origin to a pair of points which are approximately at the direction θ and the direction between the pair of points is the direction of the tangent to the boundary of the limit shape at the direction θ. The main consequence derived is the following. If σ(r) varies as rχ for some χ>0, and ξ is such that χ=2ξ−1, then the fluctuation of the transverse increment of passage time between a pair of points situated at distance r from each other is of the order of rχ∕ξ.

Universality of the geodesic tree in last passage percolation

In this paper, we consider the geodesic tree in exponential last passage percolation. We show that for a large class of initial conditions around the origin, the line-to-point geodesic that terminates in a cylinder located around the point (N,N), and whose width and length are o(N2/3) and o(N), respectively, agrees in the cylinder, with the stationary geodesic sharing the same end-point. In the case of the point-to-point model where the geodesic starts from the origin, we consider width δN2/3, length up to δ3/2N/(log(δ−1))3, and provide lower and upper bounds for the probability that the geodesics agree in that cylinder.

Hausdorff dimensions for shared endpoints of disjoint geodesics in the directed landscape

Within the Kardar–Parisi–Zhang universality class, the space-time Airy sheet is conjectured to be the canonical scaling limit for last passage percolation models. In recent work [27] of Dauvergne, Ortmann, and Virág, this object was constructed and, upon a parabolic correction, shown to be the limit of one such model: Brownian last passage percolation. The limit object without parabolic correction, called the directed landscape, admits geodesic paths between any two space-time points (x,s) and (y,t) with s<t. In this article, we examine fractal properties of the set of these paths. Our main results concern exceptional endpoints admitting disjoint geodesics. First, we fix two distinct starting locations x1 and x2, and consider geodesics traveling (x1,0)→(y,1) and (x2,0)→(y,1). We prove that the set of y∈R for which these geodesics coalesce only at time 1 has Hausdorff dimension one-half. Second, we consider endpoints (x,0) and (y,1) between which there exist two geodesics intersecting only at times 0 and 1. We prove that the set of such (x,y)∈R2 also has Hausdorff dimension one-half. The proofs require several inputs of independent interest, including (i) connections to the so-called difference weight profile studied in [10]; and (ii) a tail estimate on the number of disjoint geodesics starting and ending in small intervals. The latter result extends the analogous estimate proved for the prelimiting model in [40].

RSK in last passage percolation: a unified approach

We present a version of the RSK correspondence based on the Pitman transform and geometric considerations. This version unifies ordinary RSK, dual RSK and continuous RSK. We show that this version is both a bijection and an isometry, two crucial properties for taking limits of last passage percolation models. We use the bijective property to give a non-computational proof that dual RSK maps Bernoulli walks to nonintersecting Bernoulli walks.

Undirected Polymers in Random Environment: path properties in the mean field limit.

We consider the problem of undirected polymers (tied at the endpoints) in random environment, also known as the unoriented first passage percolation on the hypercube, in the limit of large dimensions. By means of the multiscale refinement of the second moment method we obtain a fairly precise geometrical description of optimal paths, i.e. of polymers with minimal energy. The picture which emerges can be loosely summarized as follows. The energy of the polymer is, to first approximation, uniformly spread along the strand. The polymer's bonds carry however a lower energy than in the directed setting, and are reached through the following geometrical evolution. Close to the origin, the polymer proceeds in oriented fashion -- it is thus as stretched as possible. The tension of the strand decreases however gradually, with the polymer allowing for more and more backsteps as it enters the core of the hypercube. Backsteps, although increasing the length of the strand, allow the polymer to connect reservoirs of energetically favorable edges which are otherwise unattainable in a fully directed regime. These reservoirs lie at mesoscopic distance apart, but in virtue of the high dimensional nature of the ambient space, the polymer manages to connect them through approximate geodesics with respect to the Hamming metric: this is the key strategy which leads to an optimal energy/entropy balance. Around halfway, the mirror picture sets in: the polymer tension gradually builds up again, until full orientedness close to the endpoint. The approach yields, as a corollary, a constructive proof of the result by Martinsson [Ann. Appl. Prob. 26 (2016), Ann. Prob. 46 (2018)] concerning the leading order of the ground state.

Coexistence of competing first passage percolation on hyperbolic graphs

Nous étudions un processus de croissance naturel avec compétition, qui a été récemment introduit pour analyser le MDLA, un modèle bien connu et difficile de croissance d’un agrégat par des particules diffusives. Le modèle de croissance considéré est constitué de deux processus de percolation de premier passage FPP1 et FPPλ, se propageant à taux 1 et λ>0 respectivement, sur un graphe G. Le processus FPP1 débute par un simple sommet à l’origine, alors que la condition initiale de FPPλ consiste en une infinité de germes distribués selon un produit de lois de Bernoulli de paramètre μ>0 sur V(G)∖{o}. Le processus FPP1 débute au temps 0 alors que les germes de FPPλ ne se propagent qu’après avoir été atteints par FPP1 ou FPPλ. Une question fondamentale dans ce modèle, et dans les modèles de croissance en compétition en général, est savoir si les deux processus peuvent coexister (c’est-à-dire que les deux produisent des composantes infinies) avec probabilité strictement positive. Nous montrons que c’est le cas lorsque G est transitif sur les arêtes, non-moyennable et hyperbolique, en particulier, pour tout λ>0 il existe μ0=μ0(G,λ)>0 tel que pour tout μ∈(0,μ0) les deux processus coexistent avec probabilité strictement positive. C’est la première fois qu’un résultat de coexistence non trivial est établi pour ce modèle. Nous montrons aussi que FPPλ produit une composante infinie presque surement pour tout λ, μ positifs, montrant des différences fondamentales avec le comportement de tels processus sur Zd.

Last passage percolation in an exponential environment with discontinuous rates

Last passage percolation in an exponential environment with discontinuous rates

The distance exponent for Liouville first passage percolation is positive

Discrete Liouville first passage percolation (LFPP) with parameter \(\xi > 0\) is the random metric on a sub-graph of \(\mathbb Z^2\) obtained by assigning each vertex z a weight of \(e^{\xi h(z)}\), where h is the discrete Gaussian free field. We show that the distance exponent for discrete LFPP is strictly positive for all \(\xi > 0\). More precisely, the discrete LFPP distance between the inner and outer boundaries of a discrete annulus of size \(2^n\) is typically at least \(2^{\alpha n}\) for an exponent \(\alpha > 0\) depending on \(\xi \). This is a crucial input in the proof that LFPP admits non-trivial subsequential scaling limits for all \(\xi > 0\) and also has theoretical implications for the study of distances in Liouville quantum gravity.

The oriented swap process and last passage percolation

AbstractWe present new probabilistic and combinatorial identities relating three random processes: the oriented swap process (OSP) on n particles, the corner growth process, and the last passage percolation (LPP) model. We prove one of the probabilistic identities, relating a random vector of LPP times to its dual, using the duality between the Robinson–Schensted–Knuth and Burge correspondences. A second probabilistic identity, relating those two vectors to a vector of “last swap times” in the OSP, is conjectural. We give a computer‐assisted proof of this identity for after first reformulating it as a purely combinatorial identity, and discuss its relation to the Edelman–Greene correspondence. The conjectural identity provides precise finite‐n and asymptotic predictions on the distribution of the absorbing time of the OSP, thus conditionally solving an open problem posed by Angel, Holroyd, and Romik.

Symmetries of stochastic colored vertex models

We discover a new property of the stochastic colored six-vertex model called flip-invariance. We use it to show that for a given collection of observables of the model, any transformation that preserves the distribution of each individual observable also preserves their joint distribution. This generalizes recent shift-invariance results of Borodin–Gorin–Wheeler. As limiting cases, we obtain similar statements for the Brownian last passage percolation, the Kardar–Parisi–Zhang equation, the Airy sheet and directed polymers. Our proof relies on an equivalence between the stochastic colored six-vertex model and the Yang–Baxter basis of the Hecke algebra. We conclude by discussing the relationship of the model with Kazhdan–Lusztig polynomials and positroid varieties in the Grassmannian.

Upper Tail Large Deviations in First Passage Percolation

For first passage percolation on with i.i.d. bounded edge weights, we consider the upper tail large deviation event, i.e., the rare situation where the first passage time between two points at distance n is macroscopically larger than typical. It was shown by Kesten [24] that the probability of this event decays as . However, the question of existence of the rate function, i.e., whether the log‐probability normalized by n2 tends to a limit, remains open. We show that under some additional mild regularity assumption on the passage time distribution, the rate function for upper tail large deviation indeed exists. The key intuition behind the proof is that a limiting metric structure that is atypical causes the upper tail large deviation event. The formal argument then relies on an approximate version of the above which allows us to use independent copies of the large deviation environment at a given scale to form an environment at a larger scale satisfying the large deviation event. Using this, we compare the upper tail probabilities for various values of n. © 2021 Wiley Periodicals LLC.

Bulk properties of the Airy line ensemble

The Airy line ensemble is a central object in random matrix theory and last passage percolation defined by a determinantal formula. The goal of this paper is to provide a set of tools which allow for precise probabilistic analysis of the Airy line ensemble. The two main theorems are a representation in terms of independent Brownian bridges connecting a fine grid of points, and a modulus of continuity result for all lines. Along the way, we give tail bounds and moduli of continuity for nonintersecting Brownian ensembles, and a quick proof of tightness for Dyson's Brownian motion converging to the Airy line ensemble.

THE NEAREST UNVISITED VERTEX WALK ON RANDOM GRAPHS

We revisit an old topic in algorithms, the deterministic walk on a finite graph which always moves toward the nearest unvisited vertex until every vertex is visited. There is an elementary connection between this cover time and ball-covering (metric entropy) measures. For some familiar models of random graphs, this connection allows the order of magnitude of the cover time to be deduced from first passage percolation estimates. Establishing sharper results seems a challenging problem.