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We study Busemann functions, semi-infinite geodesics, and competition interfaces in the exactly solvable last-passage percolation with inhomogeneous exponential weights. New phenomena concerning geodesics arise due to inhomogeneity. These include novel Busemann functions associated with flat regions of the limit shape and thin rectangles, undirected semi-infinite geodesics, non-trivial axis-directed geodesics, intervals with no geodesic directions, and isolated geodesic directions. We further observe a new dichotomy for competition interfaces and second-class customers in a series of memoryless continuous-time queues with inhomogeneous service rates: a second class customer either becomes trapped or proceeds through the queue at strictly positive speed., Comment: 64 pages, 4 figures

This paper studies the large scale limits of multi-type invariant distributions and Busemann functions of planar stochastic growth models in the Kardar-Parisi-Zhang (KPZ) class. We identify a set of sufficient hypotheses for convergence of multi-type invariant measures of last-passage percolation (LPP) models to the stationary horizon (SH), which is the unique multi-type stationary measure of the KPZ fixed point. Our limit theorem utilizes conditions that are expected to hold broadly in the Kardar-Parisi-Zhang class, including convergence of the scaled last-passage process to the directed landscape. We verify these conditions for the six exactly solvable models whose scaled bulk versions converge to the directed landscape, as shown by Dauvergne and Vir\'ag. We also present a second, more general, convergence theorem with potential future applications to polymer models and particle systems. Our paper is the first to show convergence to the SH without relying on information about the structure of the multi-type invariant measures of the prelimit models. These results are consistent with the conjecture that the SH is the universal scaling limit of multi-type invariant measures in the KPZ class., Comment: Final version. To appear in Int. Math. Res. Not. IMRN

We give an explicit description of the jointly invariant measures for the KPZ equation. These are couplings of Brownian motions with drift, and can be extended to a process defined for all drift parameters simultaneously. We term this process the KPZ horizon (KPZH). As a corollary of this description, we resolve a recent conjecture of Janjigian, and the second and third authors by showing the existence of a random, countably infinite dense set of directions at which the Busemann process of the KPZ equation is discontinuous. This signals instability and shows the failure of the one force--one solution principle and the existence of at least two extremal semi-infinite polymer measures in the exceptional directions. As the inverse temperature parameter $\beta$ for the KPZ equation goes to $\infty$, the KPZH converges to the stationary horizon (SH) first introduced by Busani, and studied further by Busani and the third and fourth authors. As $\beta \searrow 0$, the KPZH converges to a coupling of Brownian motions that differ by linear shifts, which is a jointly invariant measure for the Edwards-Wilkinson fixed point., Comment: v2: minor changes, 42 pages, 3 figures

We study the Busemann process of the planar directed polymer model with i.i.d. weights on the vertices of the planar square lattice, both the general case and the solvable inverse-gamma case. We demonstrate that the Busemann process intertwines with an evolution obeying a version of the geometric Robinson--Schensted--Knuth correspondence. In the inverse-gamma case this relationship enables an explicit description of the distribution of the Busemann process: the Busemann function on a nearest-neighbor edge has independent increments in the direction variable, and its distribution comes from an inhomogeneous planar Poisson process. Various corollaries follow, including that each Busemann function has the same countably infinite dense set of discontinuities in the direction variable. This contrasts with the known zero-temperature last-passage percolation cases, where the analogous sets are nowhere dense but have a dense union. The distribution of the asymptotic competition interface direction of the inverse-gamma polymer is discrete and supported on the Busemann discontinuities. Further implications follow for the eternal solutions and the failure of the one force--one solution principle for the discrete stochastic heat equation solved by the polymer partition function., Comment: 79 pages. Theorem 3.3 in this version is new

Understanding the decay of correlations in time for (1+1)-dimensional polymer models in the KPZ universality class has been a challenging topic. Following numerical studies by physicists, concrete conjectures were formulated by Ferrari and Spohn (Ferrari-Spohn '16) in the context of planar exponential last passage percolation. These have mostly been resolved by various authors. In the context of positive temperature lattice models, however, these questions have remained open. We consider the time correlation problem for the exactly solvable inverse-gamma polymer in $\mathbb{Z}^2$. We establish, up to constant factors, upper and lower bounds on the correlation between free energy functions for two polymers rooted at the origin (droplet initial condition) when the endpoints are either close together or far apart. We find the same exponents as predicted in (Ferrari-Spohn '16). Our arguments rely on the understanding of stationary polymers, coupling, and random walk comparison. We use recently established moderate deviation estimates for the free energy. In particular, we do not require asymptotic analysis of complicated exact formulae., Comment: 68 pages, 7 figures, updated references in V2

We show that two semi-infinite positive temperature polymers coalesce on the scale predicted by KPZ (Kardar-Parisi-Zhang) universality. The two polymer paths have the same asymptotic direction and evolve in the same environment, independently until coalescence. If they start at distance $k$ apart, their coalescence occurs on the scale $k^{3/2}$. It follows that the total variation distance of two semi-infinite polymer measures decays on this same scale. Our results are upper and lower bounds on probabilities and expectations that match, up to constant factors and occasional logarithmic corrections. Our proofs are done in the context of the solvable inverse-gamma polymer model, but without appeal to integrable probability. With minor modifications, our proofs give also bounds on transversal fluctuations of the polymer path. Since the free energy of a directed polymer is a discretization of a stochastically forced viscous Hamilton-Jacobi equation, our results suggest that the hyperbolicity phenomenon of such equations obeys the KPZ exponent., Comment: 48 pages, 16 figures

The Kardar-Parisi-Zhang (KPZ) equation on the real line is well-known to admit Brownian motion with a linear drift as a stationary distribution (modulo additive constants). We show that these solutions are attractive, a result known as a one force--one solution (1F1S) principle or synchronization: the solution to the KPZ equation started in the distant past from an initial condition with a given slope will converge almost surely to a Brownian motion with that drift, which shows in particular that these invariant measures are totally ergodic. Our proof constructs the Busemann process for the equation, which gives the natural jointly stationary coupling of all of these stationary solutions. Synchronization then holds simultaneously (on a single full probability event) for all but an at most countable random set of asymptotic slopes. This set of exceptional slopes of instability for which synchronization fails is either almost surely empty or almost surely dense. Along the way, we prove a shape theorem which implies almost sure stochastic homogenization of the KPZ equation, for which the Busemann process gives the process of correctors. We also show that the forward and backward point-to-point and point-to-line continuum polymers converge to semi-infinite continuum polymers whose transitions are Doob transforms via Busemann functions of the transitions of the finite length polymers., Comment: 94 pages

We show that the multi-type stationary distribution of the totally asymmetric simple exclusion process (TASEP) scales to a nontrivial limit around the Bernoulli measure of density $1/2$. This is obtained by showing that the TASEP speed process, introduced by Amir, Angel and Valk\'o, scales around the speed $v=0$ to the stationary horizon (SH), a function-valued stochastic process recently introduced and studied by the authors, SH is believed to be the universal scaling limit of Busemann processes in the KPZ universality class. Our results add to the evidence for this universality by connecting SH with multiclass particle configurations. Previously SH has been associated with the exponential corner growth model, Brownian last-passage percolation, and the directed landscape., Comment: 41 pages

We build a regular version of the field $Z_{\beta}(t,x|s,y)$ which describes the Green's function, or fundamental solution, of the parabolic Anderson model (PAM) with white noise forcing on $\mathbb{R}^{1+1}$: $\partial_t Z_{\beta}(t,x | s,y) =$ $\frac{1}{2}\partial_{xx} Z_{\beta}(t,x|s,y) + \beta Z_{\beta}(t,x | s,y)W(t,x)$, $Z_{\beta}(s,x | s,y) = \delta(x-y)$ for all $-\infty < s \leq t < \infty$, all $x,y \in \mathbb{R}$, and all $\beta \in \mathbb{R}$ simultaneously. Through the superposition principle, our construction gives a pointwise coupling of all solutions to the PAM with initial or terminal conditions satisfying sharp growth assumptions, for all initial and terminal times. Using this coupling, we show that the PAM with a (sub-)exponentially growing initial condition admits conserved quantities given by the limits $\displaystyle \lim_{x\to \pm\infty} x^{-1}\log Z_{\beta}(t,x)$, in addition to proving many new basic properties of solutions to the PAM with general initial conditions. These properties are then connected to the existence, regularity, and continuity of the quenched continuum polymer measures. Through the polymer connection, we also show that the kernel $(x,y) \mapsto Z_{\beta}(t,x | s,y)$ is strictly totally positive for all $t>s$ and $\beta\in \mathbb{R}$., Comment: Some presentation improvements over the previous version and a few slightly strengthened results. 71 pages, 3 figures

The stationary horizon (SH) is a stochastic process of coupled Brownian motions indexed by their real-valued drifts. It was first introduced by the first author as the diffusive scaling limit of the Busemann process of exponential last-passage percolation. It was independently discovered as the Busemann process of Brownian last-passage percolation by the second and third authors. We show that SH is the unique invariant distribution and an attractor of the KPZ fixed point under conditions on the asymptotic spatial slopes. It follows that SH describes the Busemann process of the directed landscape. This gives control of semi-infinite geodesics simultaneously across all initial points and directions. The countable dense set $\Xi$ of directions of discontinuity of the Busemann process is the set of directions in which not all geodesics coalesce and in which there exist at least two distinct geodesics from each initial point. This creates two distinct families of coalescing geodesics in each $\Xi$ direction. In $\Xi$ directions, the Busemann difference profile is distributed like Brownian local time. We describe the point process of directions $\xi\in\Xi$ and spatial locations where the $\xi\pm$ Busemann functions separate., Comment: v8: Accepted version. To appear in Annals of Probability. The appendices contain some proofs that are omitted from the published version

In Brownian last-passage percolation (BLPP), the Busemann functions $\mathcal B^{\theta}(\mathbf x,\mathbf y)$ are indexed by two points $\mathbf x,\mathbf y \in \mathbb Z \times \mathbb R$, and a direction parameter $\theta > 0$. We derive the joint distribution of Busemann functions across all directions. The set of directions where the Busemann process is discontinuous, denoted $\Theta$, provides detailed information about the uniqueness and coalescence of semi-infinite geodesics. The uncountable set of initial points in BLPP gives rise to new phenomena not seen in discrete models. For example, in every direction $\theta > 0$, there exists a countably infinite set of initial points $\mathbf x$ such that there exist two $\theta$-directed geodesics that split but eventually coalesce. Further, we define the competition interface in BLPP and show that the set of initial points whose competition interface is nontrivial has Hausdorff dimension $\frac{1}{2}$. From each of these exceptional points, there exists a random direction $\theta \in \Theta$ for which there exists two $\theta$-directed semi-infinite geodesics that split immediately and never meet again. Conversely, when $\theta \in \Theta$, from every initial point $\mathbf x \in \mathbb Z \times \mathbb R$, there exists two $\theta$-directed semi-infinite geodesics that eventually separate. Whenever $\theta \notin \Theta$, all $\theta$-directed semi-infinite geodesics coalesce., Comment: Accepted version. To appear in Probability and Mathematical Physics

We develop a new probabilistic method for deriving deviation estimates in directed planar polymer and percolation models. The key estimates are for exit points of geodesics as they cross transversal down-right boundaries. These bounds are of optimal cubic-exponential order. We derive them in the context of last-passage percolation with exponential weights with near-stationary boundary conditions. As a result, the probabilistic coupling method is empowered to treat a variety of problems optimally, which could previously be achieved only via inputs from integrable probability. As applications in the bulk setting, we obtain upper bounds of cubic-exponential order for transversal fluctuations of geodesics, and cube-root upper bounds with a logarithmic correction for distributional Busemann limits and competition interface limits. Several other applications are already in the literature., Comment: v3: Added MSP copyright notice. v2: Accepted version. Two main corrections: Added to Prop. 3.6 a missing assumption that was used in the proof. (The z-parameter must be close to zeta). Also added a related assumption to Thm. 4.4(b) (previously Prop. C.1), which relies on Prop. 3.6. Further minor corrections and edits. Slightly improved Thm. 3.4 and Prop. B.6. Significantly revised introduction

We prove the existence of semi-infinite geodesics for Brownian last-passage percolation (BLPP). Specifically, on a single event of probability one, there exist semi-infinite geodesics started from every space-time point and traveling in every asymptotic direction. Properties of these geodesics include uniqueness for a fixed initial point and direction, non-uniqueness for fixed direction but random initial points, and coalescence of all geodesics traveling in a common, fixed direction. Along the way, we prove that for fixed northeast and southwest directions, there almost surely exist no bi-infinite geodesics in the given directions. The semi-infinite geodesics are constructed from Busemann functions. Our starting point is a result of Alberts, Rassoul-Agha and Simper that established Busemann functions for fixed points and directions. Out of this, we construct the global process of Busemann functions simultaneously for all initial points and directions, and then the family of semi-infinite Busemann geodesics. The uncountable space of the semi-discrete setting requires extra consideration and leads to new phenomena, compared to discrete models., Comment: split the main theorems section into three sections, now sections 3-5, and reorganized the statements of the theorems for greater clarity. Also split the appendix into four sections. Updated some figures and fixed typos

We study first-passage percolation through related optimization problems over paths of restricted length. The path length variable is in duality with a shift of the weights. This puts into a convex duality framework old observations about the convergence of the normalized Euclidean length of geodesics due to Hammersley and Welsh, Smythe and Wierman, and Kesten, and leads to new results about geodesic length and the regularity of the shape function as a function of the weight shift. For points far enough away from the origin, the ratio of the geodesic length and the $\ell^1$ distance to the endpoint is uniformly bounded away from one. The shape function is a strictly concave function of the weight shift. Atoms of the weight distribution generate singularities, that is, points of nondifferentiability, in this function. We generalize to all distributions, directions and dimensions an old singularity result of Steele and Zhang for the planar Bernoulli case. When the weight distribution has two or more atoms, a dense set of shifts produce singularities. The results come from a combination of the convex duality, the shape theorems of the different first-passage optimization problems, and modification arguments., Comment: 74 pages, 10 figures

We show that nontrivial bi-infinite polymer Gibbs measures do not exist in typical environments in the inverse-gamma (or log-gamma) directed polymer model on the planar square lattice. The precise technical result is that, except for measures supported on straight-line paths, such Gibbs measures do not exist in almost every environment when the weights are independent and identically distributed inverse-gamma random variables. The proof proceeds by showing that when two endpoints of a point-to-point polymer distribution are taken to infinity in opposite directions but not parallel to lattice directions, the midpoint of the polymer path escapes. The proof is based on couplings, planar comparison arguments, and a recently discovered joint distribution of Busemann functions., Comment: 38 pages, 5 figures

We derive a lower bound for the probability that a random walk with i.i.d.\ increments and small negative drift $\mu$ exceeds the value $x>0$ by time $N$. When the moment generating functions are bounded in an interval around the origin, this probability can be bounded below by $1-O(x|\mu| \log N)$. The approach is elementary and does not use strong approximation theorems., Comment: 17 pages, 1 figure

We study moderate deviations in the exponential corner growth model, both in the bulk setting and the increment-stationary setting. The main results are sharp right-tail bounds on the last-passage time and the exit point of the increment-stationary process. The arguments utilize calculations with the stationary version and a moment generating function identity due to E. Rains, for which we give a short probabilistic proof. As applications of the deviation bounds, we derive upper bounds on the speed of distributional convergence in the Busemann function and competition interface limits., Comment: 40 pages, 6 figures

We consider point to point last passage times to every vertex in a neighbourhood of size $\delta N^{\frac{2}{3}}$, distance $N$ away from the starting point. The increments of these last passage times in this neighbourhood are shown to be jointly equal to their stationary versions with high probability that depends on $\delta$ only. With the help of this result we show that 1) the $\text{Airy}_2$ process is locally close to a Brownian motion in total variation; 2) the tree of point to point geodesics starting from every vertex in a box of side length $\delta N^{\frac{2}{3}}$ going to a point at distance $N$ agree inside the box with the tree of infinite geodesics going in the same direction; 3) two geodesics starting from $N^{\frac{2}{3}}$ away from each other, to a point at distance $N$ will not coalesce too close to either endpoints on the macroscopic scale. Our main results rely on probabilistic methods only., Comment: 40 pages, 8 figures. A result about the regularity of the Airy_2 process was added

We establish estimates for the coalescence time of semi-infinite directed geodesics in the planar corner growth model with i.i.d. exponential weights. There are four estimates: upper and lower bounds on the probabilities of both fast and slow coalescence on the correct spatial scale with exponent $3/2$. Our proofs utilize a geodesic duality introduced by Pimentel and properties of the increment-stationary last-passage percolation process. For fast coalescence our bounds are new and they have matching optimal exponential order of magnitude. For slow coalescence we reproduce bounds proved earlier with integrable probability inputs, except that our upper bound misses the optimal order by a logarithmic factor., Comment: Fixed an error in the proof of Theorem 4.1 from the previously published version. Added acknowledgments. (34 pages, 23 figures)

This paper gives a self-contained proof of the non-existence of nontrivial bi-infinite geodesics in directed planar last-passage percolation with exponential weights. The techniques used are couplings, coarse graining, and control of geodesics through planarity and estimates derived from increment-stationary versions of the last-passage percolation process., Comment: 32 pages, 9 figures

We study the macroscopic evolution of the growing cluster in the exactly solvable corner growth model with independent exponentially distributed waiting times. The rates of the exponentials are given by an addivitely separable function of the site coordinates. When computing the growth process (last-passage times) at each site, the horizontal and vertical additive components of the rates are allowed to also vary respectively with the column and row number of that site. This setting includes several models of interest from the literature as special cases. Our main result provides simple explicit variational formulas for the a.s. first-order asymptotics of the growth process under a decay condition on the rates. Formulas of similar flavor were conjectured in arXiv:math/0004082, which we also establish. Subject to further mild conditions, we prove the existence of the limit shape and describe it explicitly. We observe that the boundary of the limit shape can develop flat segments adjacent to the axes and spikes along the axes. Furthermore, we record the formation of persistent macroscopic spikes and crevices in the cluster that are nonetheless not visible in the limit shape. As an application of the results for the growth process, we compute the flux function and limiting particle profile for the TASEP with the step initial condition and disorder in the jump rates of particles and holes. Our methodology is based on concentration bounds and estimating the boundary exit probabilities of the geodesics in the increment-stationary version of the model, with the only input from integrable probability being the distributional invariance of the last-passage times under permutations of columns and rows., Comment: 46 pages, 3 figures. Shortened, minorly revised and peer-reviewed version. Author and grant information edited. Some content from the first version left out. To appear in a slightly edited form in Electronic Journal of Probability

We consider planar directed last-passage percolation on the square lattice with general i.i.d. weights and study the geometry of the full set of semi-infinite geodesics in a typical realization of the random environment. The structure of the geodesics is studied through the properties of the Busemann functions viewed as a stochastic process indexed by the asymptotic direction. Our results are further connected to the ergodic program for and stability properties of random Hamilton-Jacobi equations. In the exactly solvable exponential model, our results specialize to give the first complete characterization of the uniqueness and coalescence structure of the entire family of semi-infinite geodesics for any model of this type. Furthermore, we compute statistics of locations of instability, where we discover an unexpected connection to simple symmetric random walk., Comment: Fixed a few typos. Page layout changed. 59 pages, 13 figures

We present a proof of the almost sure existence, uniqueness and coalescence of directed semi-infinite geodesics in planar growth models that is based on properties of an increment-stationary version of the growth process. The argument is developed in the context of the exponential corner growth model. It uses coupling, planar monotonicity, and properties of the stationary growth process to derive the existence of Busemann functions, which in turn control geodesics. This soft approach is in some situations an alternative to the much-applied 20-year-old arguments of C. Newman and co-authors. Along the way we derive some related results such as the distributional equality of the directed geodesic tree and its dual, originally due to L. Pimentel., Comment: 18 pages, 2 figures. Some reorganization and additional references

The 1+1 dimensional corner growth model with exponential weights is a centrally important exactly solvable model in the Kardar-Parisi-Zhang class of statistical mechanical models. While significant progress has been made on the fluctuations of the growing random shape, understanding of the optimal paths, or geodesics, is less developed. The Busemann function is a useful analytical tool for studying geodesics. This paper describes the joint distribution of the Busemann functions, simultaneously in all directions of growth. As applications of this description we derive a marked point process representation for the Busemann function across a single lattice edge and calculate some marginal distributions of Busemann functions and semi-infinite geodesics., Comment: Improved exposition with figures

Random walk in a dynamic i.i.d. beta random environment, conditioned to escape at an atypical velocity, converges to a Doob transform of the original walk. The Doob-transformed environment is correlated in time, i.i.d. in space, and its marginal density function is a product of a beta density and a hypergeometric function. Under its averaged distribution the transformed walk obeys the wandering exponent 2/3 that agrees with Kardar-Parisi-Zhang universality. The harmonic function in the Doob transform comes from a Busemann-type limit and appears as an extremal in a variational problem for the quenched large deviation rate function., Comment: 47 pages, 6 figures. Some proofs were shortened with references to the concurrent paper arXiv:1711.08432

These lecture notes discuss several related features of the exactly solvable two-dimensional corner growth model with exponentially distributed weights. A key property of this model is the availability of a fairly explicit stationary version that possesses useful independence properties. With the help of couplings and estimates, we prove the existence of Busemann functions for this model, and the precise values of the longitudinal and transversal fluctuation exponents for the stationary corner growth model. The Busemann functions in turn furnish extremals for variational formulas that describe limiting shape functions., Comment: These notes were developed for the AMS Short Course on Random Growth Models (January 2017) and for the Research School on Random Structures in Statistical Mechanics and Mathematical Physics at CIRM in Marseille Luminy (March 2017). Later versions added two sections and made edits and corrections

We consider random walk with bounded jumps on a hypercubic lattice of arbitrary dimension in a dynamic random environment. The environment is temporally independent and spatially translation invariant. We study the rate functions of the level-3 averaged and quenched large deviation principles from the point of view of the particle. In the averaged case the rate function is a specific relative entropy, while in the quenched case it is a Donsker-Varadhan type relative entropy for Markov processes. We relate these entropies to each other and seek to identify the minimizers of the level-3 to level-1 contractions in both settings. Motivation for this work comes from variational descriptions of the quenched free energy of directed polymer models where the same Markov process entropy appears., Comment: 38 pages

We study the directed last-passage percolation model on the planar integer lattice with nearest-neighbor steps and general i.i.d. weights on the vertices, outside the class of exactly solvable models. In a previous paper we constructed stationary cocycles and Busemann functions for this model. Using these objects, we prove new results on the competition interface, on existence, uniqueness, and coalescence of directional semi-infinite geodesics, and on nonexistence of doubly infinite geodesics., Comment: 32 pages, 1 figure. Some additions and corrections. Appendix A on coalescence of cocycle geodesics has been omitted but can be found in version 1 of this submission. To appear in Probability Theory and Related Fields

We study the directed last-passage percolation model on the planar square lattice with nearest-neighbor steps and general i.i.d. weights on the vertices, outside of the class of exactly solvable models. Stationary cocycles are constructed for this percolation model from queueing fixed points. These cocycles serve as boundary conditions for stationary last-passage percolation, solve variational formulas that characterize limit shapes, and yield existence of Busemann functions in directions where the shape has some regularity. In a sequel to this paper the cocycles will be used to prove results about semi-infinite geodesics and the competition interface., Comment: 44 pages, 4 figures. Some additions and corrections. To appear in Probability Theory and Related Fields

We study the invariant distributions of Hammersley's serial harness process in all dimensions and height fluctuations in one dimension. Subject to mild moment assumptions there is essentially one unique invariant distribution, and all other invariant distributions are obtained by adding harmonic functions of the averaging kernel. We identify one Gaussian case where the invariant distribution is i.i.d. Height fluctuations in one dimension obey the stochastic heat equation with additive noise (Edwards-Wilkinson universality). We prove this for correlated initial data subject to polynomial decay of strong mixing coefficients, including process-level tightness in the Skorohod space of space-time trajectories., Comment: 40 pages

We give two variational formulas (qVar1) and (qVar2) for the quenched free energy of a random walk in random potential (RWRP) when (i) the underlying walk is directed or undirected, (ii) the environment is stationary and ergodic, and (iii) the potential is allowed to depend on the next step of the walk which covers random walk in random environment (RWRE). In the directed i.i.d. case, we also give two variational formulas (aVar1) and (aVar2) for the annealed free energy of RWRP. These four formulas are the same except that they involve infima over different sets, and the first two are modified versions of a previously known variational formula (qVar0) for which we provide a short alternative proof. Then, we show that (qVar0) always has a minimizer, (aVar2) never has any minimizers unless the RWRP is an RWRE, and (aVar1) has a minimizer if and only if the RWRP is in the weak disorder regime. In the latter case, the minimizer of (aVar1) is unique and it is also the unique minimizer of (qVar1), but (qVar2) has no minimizers except for RWRE. In the case of strong disorder, we give a sufficient condition for the nonexistence of minimizers of (qVar1) and (qVar2) which is satisfied for the log-gamma directed polymer with a sufficiently small parameter. We end with a conjecture which implies that (qVar1) and (qVar2) have no minimizers under very strong disorder., Comment: Published at http://dx.doi.org/10.3150/15-BEJ747 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

We introduce the strict-weak polymer model, and show the KPZ universality of the free energy fluctuation of this model for a certain range of parameters. Our proof relies on the observation that the discrete time geometric q-TASEP model, studied earlier by A. Borodin and I. Corwin, scales to this polymer model in the limit q->1. This allows us to exploit the exact results for geometric q-TASEP to derive a Fredholm determinant formula for the strict-weak polymer, and in turn perform rigorous asymptotic analysis to show KPZ scaling and GUE Tracy-Widom limit for the free energy fluctuations. We also derive moments formulae for the polymer partition function directly by Bethe ansatz, and identify the limit of the free energy using a stationary version of the polymer model., Comment: 23 pages, 4 figures

We study the directed last-passage percolation model on the planar integer lattice with nearest-neighbor steps and general i.i.d. weights on the vertices, outside the class of exactly solvable models. Stationary cocycles are constructed for this percolation model from queueing fixed points. These cocycles serve as boundary conditions for stationary last-passage percolation, define solutions to variational formulas that characterize limit shapes, and yield new results for Busemann functions, geodesics and the competition interface., Comment: 69 pages, 10 figures. This submission has been superseded by arXiv:1510.00859 and arXiv:1510.00860 and is no longer updated

We compute the fluctuation exponents for a solvable model of one-dimensional directed polymers in random environment in the intermediate regime. This regime corresponds to taking the inverse temperature to zero with the size of the system. The exponents satisfy the KPZ scaling relation and coincide with physical predictions. In the critical case, we recover the fluctuation exponents of the Cole-Hopf solution of the KPZ equation in equilibrium and close to equilibrium., Comment: 30 pages

We discuss variational formulas for the limits of certain models of motion in a random medium: namely, the limiting time constant for last-passage percolation and the limiting free energy for directed polymers. The results are valid for models in arbitrary dimension, steps of the admissible paths can be general, the environment process is ergodic under shifts, and the potential accumulated along a path can depend on the environment and the next step of the path. The variational formulas come in two types: one minimizes over gradient-like cocycles, and another one maximizes over invariant measures on the space of environments and paths. Minimizing cocycles can be obtained from Busemann functions when these can be proved to exist. The results are illustrated through 1+1 dimensional exactly solvable examples, periodic examples, and polymers in weak disorder., Comment: 41 pages, 2 figures. Version accepted in Communications in Mathematical Physics

We introduce a random walk in random environment associated to an underlying directed polymer model in $1+1$ dimensions. This walk is the positive temperature counterpart of the competition interface of percolation and arises as the limit of quenched polymer measures. We prove this limit for the exactly solvable log-gamma polymer, as a consequence of almost sure limits of ratios of partition functions. These limits of ratios give the Busemann functions of the log-gamma polymer, and furnish centered cocycles that solve a variational formula for the limiting free energy. Limits of ratios of point-to-point and point-to-line partition functions manifest a duality between tilt and velocity that comes from quenched large deviations under polymer measures. In the log-gamma case, we identify a family of ergodic invariant distributions for the random walk in random environment., Comment: Published at http://dx.doi.org/10.1214/14-AOP933 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We show that the geometric lifting of the RSK correspondence introduced by A.N. Kirillov (2001) is volume preserving with respect to a natural product measure on its domain, and that the integrand in Givental's integral formula for GL(n,R)-Whittaker functions arises naturally in this context. Apart from providing further evidence that Whittaker functions are the natural analogue of Schur polynomials in this setting, our results also provide a new `combinatorial' framework for the study of random polymers. When the input matrix consists of random inverse gamma distributed weights, the probability distribution of a polymer partition function constructed from these weights can be written down explicitly in terms of Whittaker functions. Next we restrict the geometric RSK mapping to symmetric matrices and show that the volume preserving property continues to hold. We determine the probability law of the polymer partition function with inverse gamma weights that are constrained to be symmetric about the main diagonal, with an additional factor on the main diagonal. The third combinatorial mapping studied is a variant of the geometric RSK mapping for triangular arrays, which is again showed to be volume preserving. This leads to a formula for the probability distribution of a polymer model whose paths are constrained to stay below the diagonal. We also show that the analogues of the Cauchy-Littlewood identity in the setting of this paper are equivalent to a collection of Whittaker integral identities conjectured by Bump (1989) and Bump and Friedberg (1990) and proved by Stade (2001, 2002). Our approach leads to new `combinatorial' proofs and generalizations of these identities, with some restrictions on the parameters., Comment: v2: significantly extended version

We consider a random walk in a random potential on a square lattice of arbitrary dimension. The potential is a function of an ergodic environment and some steps of the walk. The potential can be unbounded, but it is subject to a moment assumption whose strictness is tied to the mixing of the environment, the best case being the i.i.d. environment. We prove that the infinite volume quenched point-to-point free energy exists and has a variational formula in terms of an entropy. We establish regularity properties of the point-to-point free energy, as a function of the potential and as a function on the convex hull of the admissible steps of the walk, and link it to the infinite volume free energy and quenched large deviations of the endpoint of the walk. One corollary is a quenched large deviation principle for random walk in an ergodic random environment, with a continuous rate function., Comment: 39 pages, 3 figures, minor typos fixed

We study right tail large deviations of the logarithm of the partition function for directed lattice paths in i.i.d. random potentials. The main purpose is the derivation of explicit formulas for the $1+1$-dimensional exactly solvable case with log-gamma distributed random weights. Along the way we establish some regularity results for this rate function for general distributions in arbitrary dimensions., Comment: Published in at http://dx.doi.org/10.1214/12-AOP768 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We establish a fundamental connection between the geometric RSK correspondence and GL(N,R)-Whittaker functions, analogous to the well known relationship between the RSK correspondence and Schur functions. This gives rise to a natural family of measures associated with GL(N,R)-Whittaker functions which are the analogues in this setting of the Schur measures on integer partitions. The corresponding analogue of the Cauchy-Littlewood identity can be seen as a generalisation of an integral identity for GL(N,R)-Whittaker functions due to Bump and Stade. As an application, we obtain an explicit integral formula for the Laplace transform of the law of the partition function associated with a one-dimensional directed polymer model with log-gamma weights recently introduced by one of the authors (TS)., Comment: 31 pages,6 figures, updated introduction

We study the motion of independent particles in a dynamical random environment on the integer lattice. The environment has a product distribution. For the multidimensional case, we characterize the class of spatially ergodic invariant measures. These invariant distributions are mixtures of inhomogeneous Poisson product measures that depend on the past of the environment. We also investigate the correlations in this measure. For dimensions one and two, we prove convergence to equilibrium from spatially ergodic initial distributions. In the one-dimensional situation we study fluctuations of the net current seen by an observer traveling at a deterministic speed. When this current is centered by its quenched mean its limit distributions are the same as for classical independent particles., Comment: Final version. 41 pages

This paper is the continuation of our earlier paper, where we proved t^{1/3}-order of current fluctuations across the characteristics in a class of one dimensional interacting systems with one conserved quantity. We also claimed two models with concave hydrodynamic flux which satisfied the assumptions which made our proof work. In the present note we show that the totally asymmetric exponential bricklayers process also satisfies these assumptions. Hence this is the first example with convex hydrodynamics of a model with t^{1/3}-order current fluctuations across the characteristics. As such, it further supports the idea of universality regarding this scaling., Comment: 34 pages, revised version after referees comments, accepted at JSP. A few explanations added, some proofs shortened in the Appendix

We study directed last passage percolation on the first quadrant of the planar square lattice whose weights have general distributions, or equivalently, ./G/1 queues in series. The service time distributions of the servers vary randomly which constitutes a random environment for the model. Equivalently, each row of the last passage model has its own randomly chosen weight distribution. We investigate the limiting time constant close to the boundary of the quadrant. Close to the y-axis, where the number of random distributions averaged over stays large, the limiting time constant takes the same universal form as in the homogeneous model. But close to the x-axis we see the effect of the tail of the distribution of the random means attached to the rows., Comment: 24 pages, this paper has been accepted for publication in Stochastic Processes and their Applications