Schedule for: 23w5075 - Random Growth Models and KPZ Universality

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

In the one dimensional asymmetric simple exclusion process (ASEP), starting from a stationary state, we study the `typical' behavior of a tagged particle, conditioned to deviate to an `atypical' position at a fixed time. Among results, an `upper tail' LDP is established for the position of the tagged particle. This is work with S.R.S. Varadhan.

We will briefly introduce a totally asymmetric exclusion process on a rooted Galton-Watson tree without leaves and then focus on the aggregated current of particles across generations. Depending on the tree structure and jump rates on each node, we find upper and lower bounds for the current of particles utilising a coupling with last passage percolation. This is used to obtain time intervals that guarantee that the current across a fixed generation jumps from 0 to linear order in that interval. This is joint work with Nina Gantert and Dominik Schmid.

A survey of three new models in the KPZ class. A work with Quastel and Ramirez, a work with Veto, and a solo work of Ransford.

The KPZ class is a very large set of 1+1 models that are meant to describe random growth interfaces. It is believed that upon scaling, the long time behavior of members in this class is universal and is described by a limiting random object, a Markov process called the KPZ fixed-point. The (one-type) stationary measures for the KPZ fixed-point as well as many models in the KPZ class are known - it is a family of distributions parametrized by some set $I_{ind}$ that depends on the model. For $k\in \mathbb{N}$ the $k$-type stationary distribution with intensities $\rho_1,\dotsc,\rho_k \in I_{ind}$ is a coupling of one-type stationary measures of indices $\rho_1,\dotsc,\rho_k$ that is stationary with respect to the model dynamics. In this talk we will present recent progress in our understanding of the multi-type stationary measures of the KPZ fixed-point as well as the scaling limit of multi-type stationary measures of two families of models in the KPZ class: metric-like models (e.g. last passage percolation) and particle systems (e.g. exclusion process). Based on joint work with Timo Seppalainen and Evan Sorensen.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Denote by K(n) the convex hull of n i.i.d. random variables uniformly distributed in a smooth d-dimensional convex set K. It is shown that the re-scaled radial and longitudinal fluctuations of the boundary of K(n) asymptotically converge to explicit limit distributions as n tends to infinity, with Tracy-Widom like tails in d = 2. Re-scaled maximal radial and longitudinal fluctuations of the boundary of K(n) asymptotically follow the Gumbel law as n tends to infinity and, in d = 2, they asymptotically exhibit (1/3, 2/3) scaling, with precise logarithmic corrections. When K is the unit disc, the radial fluctuations satisfy process level convergence. We introduce a dual space-time two parameter growth process, which for t = 1 coincides with the support function of K(n), which displays 1:2:3 scaling, and which converges to a two-parameter limit process given by the Hopf-Lax formula.

In an earlier work with Chandra, Chevyrev and Hairer [CCHS’20], we constructed the local solution to the stochastic Yang-Mills equation on 2D torus, which was shown to have gauge equivariance property and thus induces a Markov process on a singular space of gauge equivalent classes. In this talk, we discuss a more recent work with Chevyrev [CS’23], where we consider the Langevin dynamics of a large class of lattice gauge theories on 2D torus, and prove that these discrete dynamics all converge to the same limiting dynamic constructed in [CCHS’20]. Using this universality result for the dynamics, we show that the Yang-Mills measure on 2D torus is the universal limit for these lattice gauge theories. We also prove that the Yang-Mills measure is invariant under the dynamic. Our argument relies on a combination of regularity structures, lattice gauge-fixing and Uhlenbeck-type argument, and Bourgain's method for invariant measures.

In the heart of the construction of the KPZ fixed point (by Matetski-Quastel-Remenik) lies the solution of TASEP with general initial conditions in terms of a random walk hitting representation. We will present a step-by-step derivation of this via principles of the RSK correspondence. Joint work with Bisi, Liao, Saenz.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Hamilton-Jacobi PDEs with random stationary Hamiltonian functions are popular toy models for studying the dynamics of interfaces in various phenomena in physics and biology. There is a one-to-one correspondence between piecewise smooth solutions and marked tessellations. A kinetic theory can be developed to describe the dynamics of such tessellations. As an application, we find kinetically describable Gibbsian solutions for such PDEs.

I will present my recent joint work with A. Davini and E. Kosygina in which we prove homogenization for viscous Hamilton-Jacobi equations with a Hamiltonian of the form $G(p) + V (x,\omega)$ for a wide class of stationary ergodic random environments in one space dimension. The momentum part $G(p)$ of the Hamiltonian is a general (nonconvex) continuous function with superlinear growth at infinity, and the potential $V(x,\omega)$ is a bounded and Lipschitz continuous function of $x$ that does not depend on time. The class of random environments we consider is defined by an explicit hill and valley condition on the diffusivity-potential pair which holds as long as the environment is not "rigid". We show that, outside of the intervals where the graph of the effective Hamiltonian turns out to be flat, there are sublinear correctors, which correspond to Busemann functions in the context of several models of interest to this workshop where $G(p)$ is a parabola.

I will present quantitative estimates on the parabolic Green function and the stationary invariant measure in the context of stochastic homogenization of elliptic equations in nondivergence form. I will discuss implications of these homogenization results, such as a quenched, local CLT for the corresponding diffusion process and a quantitative ergodicity estimate for the environmental process. This talk is based on joint work with Scott Armstrong (NYU) and Benjamin Fehrman (Oxford).

For several classes of continuous space polymer models in positive and zero temperature, we show that the shape function also known as effective Lagrangian is differentiable and give a formula for its derivative. Joint work with Douglas Dow.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

This talk presents the convergence of the KPZ equation to the directed landscape. This convergence result is the first to the directed landscape among the positive temperature models.

Consider the geodesic in the directed landscape between (0, 0) and (0,1), and condition on the event that its weight is at least some large value $L$. It was proven by Zhipeng Liu using formulas from integrable probability that the one-point distribution of the geodesic under this conditioning, suitably rescaled, converges as $L\to\infty$ to the one-point distribution of the Brownian bridge, and he conjectured that the convergence holds as a process. We will discuss work in progress with Shirshendu Ganguly and Lingfu Zhang proving this conjecture. Our approach does not rely on exact formulas but instead makes crucial use of the parabolic Airy line ensemble, its Brownian Gibbs resampling property, and a recent description developed with Ganguly of how it looks under the upper tail conditioning.

The directed landscape is a random directed metric on the plane that is the scaling limit for models in the KPZ universality class. In this metric, typical pairs of points are connected by a unique geodesic. However, certain exceptional pairs are connected by more exotic geodesic networks. The goal of this talk is to describe a full classification for these exceptional pairs.

I will introduce a new method to construct the stationary measure for open boundary KPZ models, focusing on geometric LPP, the log-gamma polymer, and the KPZ equation. The method realizes these stationary measures as marginals of two-layer Gibbs measures by constructing local Markov dynamics that preserve this class of measures and project on the top layer to the KPZ models. This is based on joint work with Guillaume Barraquand and Zongrui Yang.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Directed polymers in dimension 2 exhibit weak disorder under appropriate scaling of the temperature. I will describe the evolution of high moments of their partition function. Based on joinbt works with Clement Cosco.

Consider the n-point, fixed-time large deviations of the Kardar--Parisi--Zhang (KPZ) equation with the narrow wedge initial condition. The scope consists of concave-configured, upper-tail deviations and a range of scaling regimes that allows time to be short, unit-order, and long. I will present a result (joint with Yier Lin) on the n-point Large Deviation Principle (LDP) and the corresponding spacetime limit shape. The proof is based on another work (of myself) on the multipoint moments of the Stochastic Heat Equation (SHE). I will explain how to analyze the moments via a system of attractive Brownian particles and how to use the moments to obtain the LDP and spacetime limit shape.

The Kardar-Parisi-Zhang (KPZ) equation on the line is known to admit stationary distributions modulo additive constants given by Brownian motion with drift. In this talk, we will discuss some recent results studying the ergodicity properties of these distributions through the lens of the synchronization by noise phenomenon. We show that the solution to the equation started in the distant past from an initial condition with a given slope will converge almost surely to a Brownian motion with that same drift for all but a random countable set of slopes. Our analysis centers on the family of Brownian motions obtained in this way, which is known as the Busemann process for the equation, and connections to the infinite volume structure of the continuum directed polymer. Based on joint works with Tom Alberts (Utah), Firas Rassoul-Agha (Utah), and Timo Seppäläinen (Madison).

We show the existence of a random, countably infinite dense set of directions for which the Busemann process for the KPZ equation is discontinuous. This shows the failure of the One Force--One Solution principle in those exceptional directions and resolves a recent conjecture of Janjigian, Rassoul-Agha, and Seppäläinen. This is proved by constructing a coupling of Brownian motions, which we term the KPZ horizon. We study the asymptotics of the KPZ horizon as the temperature goes to 0 and as temperature goes to $\infty$.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In the KMP model each vertex $i$ of a finite graph has an energy $X_i$, a nonnegative real value. When a Poisson clock rings at the edge $ij$, the current energies $X_i$ and $X_j$ are substituted by $U(X_i+X_j)$ and $(1-U)(X_i+X_j)$, respectively, where $U$ is uniform in $(0,1)$. If $j$ is a boundary vertex, it is updated to an exponential variable with mean $T_j$, a temperature parameter. We show that the invariant measure is the law of $Z$, a vector with coordinates $Z_i = T_i X_i$, where $X_i$ are iid exponential(1), the law of $T$ is the invariant measure for an opinion model with the same boundary conditions, and $X$ and $T$ are independent. The result confirms a conjecture based on the large deviations of KMP. Applications include hydrodynamic limits. The opinion differences associated to edges behave as a neural spiking process. Work in progress with Anna de Masi and Davide Gabrielli.

We will introduce a periodic version of the PNG model and show that it is a solvable model. We can give stationary measures for the model at a fixed time and for the distribution of the space-time paths, which in this model are up-down paths that form rings. This is joint work with Pablo Ferrari (UBA).

I will explain how the two-parameter stationary last passage percolation picture and some planarity tricks can be used to establish stabilisation of the point-to-point geodesic tree to the semi-infinite one. The repertoire includes Poisson processes and loaded queues. Joint work with Ofer Busani and Timo Seppäläinen.

Busemann functions have been decidedly instrumental in advancing our understanding of FPP and LPP, both in general and exactly solvable cases. The Busemann theory is less developed, however, for their positive-temperature counterparts: directed polymers. This talk will highlight some recent progress on that front, namely in understanding the Busemann process simultaneously across all directions. We will see some reassuring similarities with the zero-temperature setting, but also some intriguing differences. The inverse-gamma model will be our star witness.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will present a general methodology, based on a formula of Rains and Emrah-Janjigian-Seppalainen, to obtain scaling and tail bounds in several KPZ models, including some that are not known or expected to be integrable.

The temporal correlations of randomly growing interfaces within the KPZ universality class have gained significant interest in recent research. In this talk, we address the time correlation problem for the inverse-gamma polymer in (1+1) dimensions, examining both droplet and diffusive growth initial conditions. We establish upper and lower bounds, up to constant factors, for the correlation between the free energy of two polymers with endpoints in close proximity or far apart. Our arguments rely on the understanding of stationary polymers, coupling, random walk comparison, and the recently established one-point moderate deviation estimates derived through stationary polymer techniques. Therefore, our methods are only weakly dependent on the integrable nature of the model. (Based on joint work with Timo Seppäläinen and Riddhipratim Basu.)

A Gelfand-Tsetlin (GT) pattern of depth n is an interlacing array of $n(n+1)/2$ real entries distributed over n levels such that level $k=0, 1,\dotsc, n-1$ contains exactly $n-k$ entries. Let $G_n$ denote a GT pattern with a fixed level zero sequence $a_n$ and with the remaining entries (particles) distributed uniformly at random. By a result of Y. Baryshnikov, $G_n$ is equal in distribution to the eigenvalue minor process of a unitarily invariant random Hermitian matrix with eigenvalue sequence $a_n$. In this talk, our interest is in the limiting boundary fluctuations of $G_n$ at the level of finite-dimensional distributions of first particles. We present a classification theorem that identifies five fluctuation regimes in terms of the level zero data $(a_n)$ and describes the corresponding limit processes. This result is from a forthcoming joint work with Kurt Johansson.

The complex eigenvalues $z_1,\ldots, z_N$ of non-Hermitian random matrices $H=H_0+i\Gamma$ ($H_0$ is a Hermitian random matrix and $\Gamma$ is a deterministic symmetric matrix of the finite rank) have attracted much research interest due to their relevance to several branches of theoretical physics, and in particular to the study of scattering chaotic systems. Taken $H_0$ from the few classical ensembles of random matrices, we discuss a certain form of universality appearing in the distribution of $\Im z_1,\ldots, \Im z_N$ ($\{\Im z_i\}$ physically correspond to the so-called ``resonance widths")

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Many combinatorial games, such as chess, Go and Hex, are zero-sum games in which two players alternate in making moves. In a random turn variant, each player wins the right to move at any given turn according to the flip of a fair coin. In 2007, Peres, Schramm, Sheffield and Wilson [PSSW] found explicit optimal strategies for a broad class of random-turn games, including Hex. The gameplay in random turn Hex is a novel random growth process related to planar critical percolation; other random turn games offer growth processes related to other universality classes.

Take a random configuration of $(a,b,c)$-weighted six-vertex model in a very large planar domain. What does it look like near a straight segment of the boundary? We investigate this question on the example of the model in $N\times N$ square with Domain Wall Boundary Conditions and find that the answer depends on the value of $\Delta=(a^2+b^2-c^2)/(2ab)$: there is a single universal limiting object for all $\Delta<1$ and a richer class of limits at $\Delta>1$.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building