arXiv subjects

Ivan Corwin

Publications and source records attributed to Ivan Corwin.

At least 19 recordsLinked to original sources

arXiv2026

Periodic directed landscape

We construct the periodic directed landscape, which is the conjectured scaling limit for periodic models in the Kardar-Parisi-Zhang universality class. We establish the convergence of periodic exponential last passage percolation to the periodic directed landscape. Moreover, we confirm conjectures by Baik, Liao and Liu on the local structure of the periodic KPZ fixed point, and establish the convergence of periodic ASEPs to periodic KPZ fixed points, coupled according to the same periodic directed landscape. Our main tool to construct the periodic directed landscape (and prove convergence to it) is a technique for gluing full-space directed landscapes (and their prelimits), which is of independent interest.

math.PR↗

arXiv2025

Universal KPZ Fluctuations for Moderate Deviations of Random Walks in Random Environments

The theory of diffusion seeks to describe the motion of particles in a chaotic environment. Classical theory models individual particles as independent random walkers, effectively forgetting that particles evolve together in the same environment. Random Walks in a Random Environment (RWRE) models treat the environment as a random space-time field that biases the motion of particles based on where they are in the environment. We provide a universality result for the moderate deviations of the transition probability of this model over a wide class of choices of random environments. In particular, we show the convergence of moments to those of the multiplicative noise stochastic heat equation (SHE), whose logarithm is the Kardar-Parisi-Zhang (KPZ) equation. The environment only filters into the scaling limit through one parameter, which depends explicitly on the statistical description of the environment. This forms the basis for our introduction, in arXiv:2406.17733, of the extreme diffusion coefficient.

cond-mat.stat-mech↗

arXiv2024

KPZ fixed point convergence of the ASEP and stochastic six-vertex models

We consider the stochastic six-vertex (S6V) model and asymmetric simple exclusion process (ASEP) under general initial conditions which are bounded below lines of arbitrary slope at $\pm\infty$. We show under Kardar-Parisi-Zhang (KPZ) scaling of time, space, and fluctuations that the height functions of these models converge to the KPZ fixed point. Previously, our results were known in the case of ASEP (for a particular direction in the rarefaction fan) via a comparison approach arXiv:2008.06584.

math.PR↗

arXiv2024

Periodic Pitman transforms and jointly invariant measures

We construct explicit jointly invariant measures for the periodic KPZ equation (and therefore also the stochastic Burgers' and stochastic heat equations) for general slope parameters and prove their uniqueness via a one force--one solution principle. The measures are given by polymer-like transforms of independent Brownian bridges. We describe several properties and limits of these measures, including an extension to a continuous process in the slope parameter that we term the periodic KPZ horizon. As an application of our construction, we prove a Gaussian process limit theorem with an explicit covariance function for the long-time height function fluctuations of the periodic KPZ equation when started from varying slopes. In connection with this, we conjecture a formula for the fluctuations of cumulants of the endpoint distribution for the periodic continuum directed random polymer. To prove joint invariance, we address the analogous problem for a semi-discrete system of SDEs related to the periodic O'Connell-Yor polymer model and then perform a scaling limit of the model and jointly invariant measures. For the semi-discrete system, we demonstrate a bijection that maps our systems of SDEs to another system with product invariant measure. Inverting the map on this product measure yields our invariant measures. This map relates to a periodic version of the discrete geometric Pitman transform that we introduce and probe. As a by-product of this, we show that the jointly invariant measures for a periodic version of the inverse-gamma polymer are the same as those for the O'Connell-Yor polymer.

math.PR↗

arXiv2024

Extreme Diffusion Measures Statistical Fluctuations of the Environment

We consider many-particle diffusion in one spatial dimension modeled as Random Walks in a Random Environment (RWRE). A shared short-range space-time random environment determines the jump distributions that drive the motion of the particles. We determine universal power-laws for the environment's contribution to the variance of the extreme first passage time and extreme location. We show that the prefactors rely upon a single extreme diffusion coefficient that is equal to the ensemble variance of the local drift imposed on particles by the random environment. This coefficient should be contrasted with the Einstein diffusion coefficient, which determines the prefactor in the power-law describing the variance of a single diffusing particle and is equal to the jump variance in the ensemble averaged random environment. Thus a measurement of the behavior of extremes in many-particle diffusion yields an otherwise difficult to measure statistical property of the fluctuations of the generally hidden environment in which that diffusion occurs. We verify our theory and the universal behavior numerically over many RWRE models and system sizes.

cond-mat.stat-mech↗

arXiv2024

Scaling limit of the colored ASEP and stochastic six-vertex models

We consider the colored asymmetric simple exclusion process (ASEP) and stochastic six vertex (S6V) model with fully packed initial conditions; the states of these models can be encoded by 2-parameter height functions. We show under Kardar-Parisi-Zhang (KPZ) scaling of time, space, and fluctuations that these height functions converge to the Airy sheet. Several corollaries follow. (1) For ASEP and the S6V model under the basic coupling, we consider the 4-parameter height function at position $y$ and time $t$ with a step initial condition at position $x$ and time $s < t$, and prove that under KPZ scaling it converges to the directed landscape. (2) We prove that ASEPs under the basic coupling, with multiple general initial data, converge to KPZ fixed points coupled through the directed landscape. (3) We prove that the colored ASEP stationary measures converge to the stationary horizon. (4) We prove a strong form of decoupling for the colored ASEP height functions, as well as for the stationary two-point function, as broadly predicted by the theory of non-linear fluctuating hydrodynamics. The starting point for our Airy sheet convergence result is an embedding of these colored models into a larger structure -- a color-indexed family of coupled line ensembles with an explicit Gibbs property, i.e., a colored Hall-Littlewood line ensemble. The core of our work then becomes to develop a framework to analyze the edge scaling limit of these ensembles.

math.PR↗

arXiv2023

KPZ exponents for the half-space log-gamma polymer

We consider the point-to-point log-gamma polymer of length $2N$ in a half-space with i.i.d. $\operatorname{Gamma}^{-1}(2\theta)$ distributed bulk weights and i.i.d. $\operatorname{Gamma}^{-1}(\alpha+\theta)$ distributed boundary weights for $\theta>0$ and $\alpha>-\theta$. We establish the KPZ exponents ($1/3$ fluctuation and $2/3$ transversal) for this model when $\alpha=N^{-1/3}\mu$ for $\mu\in \mathbb{R}$ fixed (critical regime) and when $\alpha>0$ is fixed (supercritical regime). In particular, in these two regimes, we show that after appropriate centering, the free energy process with spatial coordinate scaled by $N^{2/3}$ and fluctuations scaled by $N^{1/3}$ is tight. These regimes correspond to a polymer measure which is not pinned at the boundary. This is the first instance of establishing the $2/3$ transversal exponent for a positive temperature half-space model, and the first instance of the $1/3$ fluctuation exponent besides precisely at the boundary where recent work of arXiv:2204.08420 applies and also gives the exact one-point fluctuation distribution (our methods do not access exact fluctuation distributions). Our proof relies on two inputs -- the relationship between the half-space log-gamma polymer and half-space Whittaker process (facilitated by the geometric RSK correspondence as initiated in arXiv:1110.3489, arXiv:1210.5126), and an identity in arXiv:2108.08737 which relates the point-to-line half-space partition function to the full-space partition function for the log-gamma polymer. The primary technical contribution of our work is to construct the half-space log-gamma Gibbsian line ensemble and develop, in the spirit of work initiated in arXiv:1108.2291, a toolbox for extracting tightness and absolute continuity results from minimal information about the top curve of such half-space line ensembles. This is the first study of half-space line ensembles.

math.PR↗

arXiv2023

First Passage Time for Many Particle Diffusion in Space-Time Random Environments

The first passage time for a single diffusing particle has been studied extensively, but the first passage time of a system of many diffusing particles, as is often the case in physical systems, has received little attention until recently. We consider two models for many particle diffusion -- one treats each particle as independent simple random walkers while the other treats them as coupled to a common space-time random forcing field that biases particles nearby in space and time in similar ways. The first passage time of a single diffusing particle under both of these models show the same statistics and scaling behavior. However, for many particle diffusions, the first passage time among all particles (the `extreme first passage time') is very different between the two models, effected in the latter case by the randomness of the common forcing field. We develop an asymptotic (in the number of particles and location where first passage is being probed) theoretical framework to separate out the impact of the random environment with that of sampling trajectories within it. We identify a new power-law describing the impact to the extreme first passage time variance of the environment. Through numerical simulations we verify that the predictions from this asymptotic theory hold even for systems with widely varying numbers of particles, all the way down to 100 particles. This shows that measurements of the extreme first passage time for many-particle diffusions provide an indirect measurement of the underlying environment in which the diffusion is occurring.

cond-mat.stat-mech↗

arXiv2023

Francis Comets' Gumbel last passage percolation

In 2015, Francis Comets shared with me a clever way to relate a model of directed last passage percolation with i.i.d. Gumbel edge weights to a special case of the log-gamma directed polymer model. To my knowledge, he never wrote this down. In the wake of his recent passing I am recording Francis' observation along with some associated asymptotics and discussion. This note is dedicated in memory of Francis whose work in the study of directed polymers defined and refined the field immensely.

math.PR↗

arXiv2023

Stationary measures for integrable polymers on a strip

We prove that the stationary measures for the free-energy increment process for the geometric last passage percolation (LPP) and log-gamma polymer model on a diagonal strip is given by a marginal of a two-layer Gibbs measure with a simple and explicit description. This result is shown subject to certain restrictions on the parameters controlling the weights on the boundary of the strip. However, from this description and an analytic continuation argument we are able to access the stationary measure for all boundary parameters. Taking an intermediate disorder limit of the log-gamma polymer stationary measure in a strip we readily recover (modulo convergence of the polymer to the open KPZ equation, Conjecture 4.2) the conjectural description from arXiv:2105.15178 of the open KPZ stationary measure for all choices of boundary parameters $u,v\in \mathbb{R}$ (thus going beyond the restriction $u+v\geq 0$ from arXiv:2103.12253).

math.PR↗

arXiv2022

Markov duality and Bethe ansatz formula for half-line open ASEP

Using a Markov duality satisfied by ASEP on the integer line, we deduce a similar Markov duality for half-line open ASEP and open ASEP on a segment. This leads to closed systems of ODEs characterizing observables of the models. In the half-line case, we solve the system of ODEs using Bethe ansatz and prove an integral formula for $q$-moments of the current at $n$ distinct spatial locations. We then use this formula to confirm predictions for the moments of the multiplicative noise stochastic heat equation on $\mathbb R_{>0}$ with Robin type boundary condition and we obtain new formulas in the case of a Dirichlet boundary condition.

math.PR↗

arXiv2022

The lower tail of $q$-pushTASEP

We study $q$-pushTASEP, a discrete time interacting particle system whose distribution is related to the $q$-Whittaker measure. We prove a uniform in $N$ lower tail bound on the fluctuation scale for the location $x_N(N)$ of the right-most particle at time $N$ when started from step initial condition. Our argument relies on a map from the $q$-Whittaker measure to a model of periodic last passage percolation (LPP) with geometric weights in an infinite strip that was recently established in [arXiv:2106.11922]. By a path routing argument we bound the passage time in the periodic environment in terms of an infinite sum of independent passage times for standard LPP on $N\times N$ squares with geometric weights whose parameters decay geometrically. To prove our tail bound result we combine this reduction with a concentration inequality, and a crucial new technical result -- lower tail bounds on $N\times N$ last passage times uniformly over all $N \in \mathbb N$ and all the geometric parameters in $(0,1)$. This technical result uses Widom's trick [arXiv:math/0108008] and an adaptation of an idea of Ledoux introduced for the GUE [Led05a] to reduce the uniform lower tail bound to uniform asymptotics for very high moments, up to order $N$, of the Meixner ensemble. This we accomplish by first obtaining sharp uniform estimates for factorial moments of the Meixner ensemble from an explicit combinatorial formula of Ledoux [Led05b], and translating them to polynomial bounds via a further careful analysis and delicate cancellation.

math.PR↗

arXiv2022

The ASEP speed process

For ASEP with step initial data and a second class particle started at the origin we prove that as time goes to infinity the second class particle almost surely achieves a velocity that is uniformly distributed on $[-1,1]$. This positively resolves Conjecture 1.9 and 1.10 of [Amir, Angel and Valko, "The TASEP speed process", Annals of Probability 39, 1205--1242, 2011] and allows us to construct the ASEP speed process.

math.PR↗

arXiv2022

Stationary measures for the log-gamma polymer and KPZ equation in half-space

We construct explicit one-parameter families of stationary measures for the Kardar-Parisi-Zhang equation in half-space with Neumann boundary conditions at the origin, as well as for the log-gamma polymer model in a half-space. The stationary measures are stochastic processes that depend on the boundary condition as well as a parameter related to the drift at infinity. They are expressed in terms of exponential functions of Brownian motions and gamma random walks. We conjecture that these constitute all extremal stationary measures for these models. The log-gamma polymer result is proved through a symmetry argument related to half-space Whittaker processes which we expect may be applicable to other integrable models. The KPZ result comes as an intermediate disorder limit of the log-gamma polymer result and confirms the conjectural description of these stationary measures from arXiv:2105.15178. To prove the intermediate disorder limit, we provide a general half-space polymer convergence framework that extends works of arXiv:1804.09815, arXiv:1901.09449, arXiv:1202.4398.

math.PR↗

arXiv2022

Some recent progress on the stationary measure for the open KPZ equation

This note, dedicated in Harold Widom's memory, is an expanded version of a lecture I gave in fall 2021 at the MSRI program "Universality and Integrability in Random Matrices and Interacting Particle Systems". I will focus on the behavior of the stationary measure for the open KPZ equation, a paradigmatic model for interface growth in contact with boundaries. Much of this will review elements of my joint work with A. Knizel as well as with H. Shen, as well as subsequent works of W. Bryc, A. Kuznetsov, Y. Wang, and J. Wesolowski and of G. Barraquand and P. Le Doussal. The basis for this advance is fundamental work of B. Derrida, M. Evans, V. Hakim and V. Pasquier from 1993, of T. Sasamoto, M. Uchiyama and M. Wadati from 2003, and of W. Bryc and J. Wesolowski from 2010 and 2017. I will try to explain how all of this fits together, without laboring details for the sake of exposition. Though this work does not directly follow from Harold Widom's own work, it (and a great deal of my research) is very much inspired by his and Craig Tracy's work on ASEP.

math.PR↗

arXiv2022

Harold Widom's work in random matrix theory

This is a survey of Harold Widom's work in random matrices. We start with his pioneering papers on the sine-kernel determinant, continue with his and Craig Tracy's groundbreaking results concerning the distribution functions of random matrix theory, touch on the remarkable universality of the Tracy-Widom distributions in mathematics and physics, and close with Tracy and Widom's remarkable work on the asymmetric simple exclusion process.

math-ph↗

arXiv2021

Maximal free energy of the log-gamma polymer

We prove a phase transition for the law of large numbers and fluctuations of $\mathsf F_N$, the maximum of the free energy of the log-gamma directed polymer with parameter $\theta$, maximized over all possible starting and ending points in an $N\times N$ square. In particular, we find an explicit critical value $\theta_c=2\Psi^{-1}(0)>0$ ($\Psi$ is the digamma function) such that: 1. For $\theta<\theta_c$, $\mathsf F_N+2\Psi(\theta/2)N$ has order $N^{1/3}$ GUE Tracy-Widom fluctuations. 2. For $\theta=\theta_c$, $\mathsf F_N= \Theta(N^{1/3}(\log N)^{2/3})$. 3. For $\theta>\theta_c$, $\mathsf F_N=\Theta(\log N)$. Using a connection between the log-gamma polymer and a certain random operator on the honeycomb lattice, recently found by Kotowski and Vir\'ag (Commun. Math. Phys. 370, 2019), we deduce a similar phase transition for the asymptotic behavior of the smallest positive eigenvalue of the aforementioned random operator.

math.PR↗

arXiv2021

Stationary measure for the open KPZ equation

We provide the first construction of stationary measures for the open KPZ equation on the spatial interval $[0,1]$ with general inhomogeneous Neumann boundary conditions at $0$ and $1$ depending on real parameters $u$ and $v$, respectively. When $u+v\geq 0$ we uniquely characterize the constructed stationary measures through their multipoint Laplace transform which we prove is given in terms of a stochastic process that we call the continuous dual Hahn process.

math.PR↗