Titles and Abstracts
Martin Hairer: Supercritical KPZ equations with long-range noise
Yu Gu: Effective diffusivities of periodic KPZ
It’s known that the KPZ equation on a ring exhibits diffusive behaviors in large time, and in this talk I will present the recent joint work with Alex Dunlap and Tomasz Komorowski in which we derived nearly explicit expressions of the diffusion constants.
Li-Cheng Tsai: Hydrodynamic large deviations of TASEP
Consider the large deviations from the hydrodynamic limit of the Totally Asymmetric Simple Exclusion Process (TASEP), which is related to the entropy production in the inviscid Burgers equation. I will present a result, jointly with Jeremy Quastel, on the full large deviation principle. Our method relies on the explicit formula of Matetski, Quastel, and Remenik (2016) for the transition probabilities of the TASEP.
Amol Aggarwal: Scaling limit of the Colored ASEP
In this talk we consider the colored, also called multi-species, asymmetric simple exclusion process (ASEP), and review recent results showing its convergence to the Airy sheet. The proofs proceed by first using the Yang-Baxter relation to map the colored process onto a family of line ensembles that all interact with each other, and then probabilistically analyzing the latter family to identify its scaling limit.
Duncan Dauvergne: Three limits of RSK
Abstract: I will discuss three successive limits of the classical RSK correspondence that are relevant to the KPZ class. The last of these is a continuum version of the RSK correspondence in the scaling limit, which gives a way of building the directed landscape from a sequence of independent Brownian motions. Based on joint work with Balint Virág.
Timo Seppäläinen: Busemann functions and multicomponent stationary measures in stochastic growth models
This talk is an overview of the properties of multicomponent stationary measures and Busemann functions of random growth models and directed polymer models, and of the utility of these measures in the study of the properties of the models. Examples include the corner growth model, the planar directed polymer model, the KPZ equation, and the directed landscape.
Ron Peled: Minimal Surfaces in Random Environment
A minimal surface in a random environment (MSRE) is a surface which minimizes the sum of its elastic energy and its environment potential energy, subject to prescribed boundary conditions. Apart from their intrinsic interest, such surfaces are further motivated by connections with disordered spin systems, first-passage percolation models and minimal cuts in the Z^d lattice with random capacities.
We wish to study the geometry of d-dimensional minimal surfaces in a (d+n)-dimensional random environment. Specializing to a model that we term harmonic MSRE, in an ``independent'' random environment, we rigorously establish bounds on the geometric and energetic fluctuations of the minimal surface, as well as a scaling relation that ties together these two types of fluctuations. In particular, we prove, for all values of n, that the surfaces are delocalized in dimensions d≤4 and localized in dimensions d≥5. Moreover, the surface delocalizes with power-law fluctuations when d≤3 and with sub-power-law fluctuations when d=4. Many of our results are new even for d=1 (indeed, even for d=n=1), where our model is a kind of (non-integrable) first-passage percolation.
Based on joint work with Barbara Dembin, Dor Elboim and Daniel Hadas. Joint work with Michal Bassan and Shoni Gilboa will also be discussed.
Kurt Johansson: Coulomb gas and the Grunsky operator on a Jordan domain with corners
I will discuss the asymptotics of the partition function of a Coulomb gas in the plane confined to a region bounded by a Jordan curve with corners. The analysis is based on an exact formula that expresses the partition function as a finite Fredholm determinant of an operator coming from the Grunsky operator for the curve. From this formula it follows immediately that the suitably normalized partition function has a limit if and only if the Jordan curve is a Weil-Petersson quasicircle. If the curve has corners it is no longer a Weil-Petersson quasi circle and the the suitably normalized partition function diverges like a power of n, the number of particles, and the exponent depends on the angles in an interesting way.
Gérard Ben Arous: The Mezard-Parisi elastic manifold
The Elastic Manifold is a fascinating classical model of a disordered elastic system. It boils down to a large lattice system of spin glasses in an elastic interaction. This model has been deeply studied in the Physics literature since the 90’s, starting with the fundamental work by Mezard and Parisi, and more recently in beautiful works by Fyodorov and Le Doussal. The basic picture emerging is naturally that the quenched disorder tends to induce complexity at low temperature, while the elastic interaction tends to tame this complexity. After quickly reviewing recent progress with Paul Bourgade (Courant) and Benjamin Mc Kenna (Harvard) about the topological complexity of this model (at zero temperature), I will report on current works with Pax Kivimae (Courant) on the positive temperature behavior. We give a full (and unusual) variational formula for the quenched free energy, and a first understanding of the low temperature phase, and in particular the strange role of the so-called Larkin mass at positive temperature.
Konstantin Matetski: Solving interacting particle systems with different speeds and memory lengths
We explain an explicit bi-orthogonalization method of solution for a class of determinantal measures, whose marginal distributions in particular cases coincide with transition probabilities of TASEP and similar models with different speeds and memory lengths. We obtain a new spatially inhomogeneous scaling limit of TASEP with suitably modified speeds around a characteristic line. This is a joint work with Daniel Remenik.
Evita Nestoridi: Cutoff for biased transpositions
Diaconis and Shahshahani proved that shuffling a deck of n cards according with random transpositions takes $1/2 n \log n$ steps to mix. In this talk we will discuss the case where a card that is located in the top $n/2$ positions gets selected with probability $a/n$ and otherwise it gets selected with probability $(2-a)/n$, where $0<a<=1$ is fixed. We then swap the cards. In joint work in progress with A. Yan, we prove that this shuffle takes $(2a)^{-1} n \log n$ steps to mix. Our proof heavily relies on the results of Diaconis and Shahshahani for random transpositions.
Bálint Virág: A large deviation principle for the directed landscape
w Dauvergne and Das
Benjamin Landon: The maximum of the characteristic polynomial for non-Hermitian random matrices
In 2020, Lambert computed the leading order asymptotics of the logarithm of the maximum of the characteristic polynomial of the Ginibre ensemble. In this talk we discuss the extension of this result to general random matrices, not necessarily having Gaussian entries.
Herbert Spohn: Two coupled KPZ equations
Reviewed will be models with two conserved fields which have a non-degenerate flux Jacobian, thereby leading to two peaks travelling at distinct velocities. In recent work, jointly with Dhar, Khanin, Kulkarni, and Roy, studied is the case of a degenerate flux Jacobian. Novel scaling functions are revealed.
Alexander Dunlap: Stochastic heat equations and Cauchy distributions
I will describe how an invariant measure with Cauchy-distributed marginals arises from the multiplicative stochastic heat equation with an additional, independent additive noise. Joint work with Chiranjib Mukherjee.
Horng-Tzer Yau: Spectral gap and two point function estimates for mean-field spin glass models
In this lecture, we will review a method to prove that the spectral gaps of the Glauber dynamics for a class of mean-field spin glass models are of order one at sufficiently high temperatures. In addition, we will present estimates on two point functions for the SK model satisfying a modified de Almeida–Thouless condition.
Patrik Ferrari: Limit processes for flat KPZ
We will give an overview of older and recent results on the limit process and its properties for KPZ growth models with flat initial conditions, ranging from regularity, decorrelation to persistence.
Alessandra Occelli: Towards a Painlevé formula for PNG model with external sources in half-space
In this talk I will introduce a preliminary result concerning the distribution of a PNG model in half-space with two external sources. First I will present the strategy developed to study the model in the full space setting (Baik--Rains '00); strategy which relies on algebraic and orthogonal polynomials identities, and Riemann--Hilbert techniques, and which led to a limit distribution formulated in terms of the solution to Painlevé II Riemann--Hilbert problem. I will try to underline the differences and the difficulties we face when looking at the asymptotic regime in half-space.
Sourav Chatterjee: Chaos in lattice spin glasses
In spite of tremendous progress in the mean-field theory of spin glasses in the last forty years, culminating in Giorgio Parisi’s Nobel Prize in 2021, the more “realistic” short-range spin glass models have remained almost completely intractable. In this talk, I will show that the ground states of short-range spin glasses are chaotic with respect to small perturbations of the disorder, settling a conjecture made by Daniel Fisher and David Huse in 1986.
Tomohiro Sasamoto: Skew RSK dynamics and KPZ models
The Fredholm determinant formula for the KPZ equation suggests that it is related to a free fermion at finite temperature. In [1] we found a connection between discrete KPZ models and free fermions through the skew RSK dynamics, which we introduced as a time evolution for a pair of skew Young tableaux (P,Q). The dynamics exhibits solitonic behaviors similar to box ball systems (BBS) and provides a connection between the q-Whittaker measure and the periodic Schur measure.
In this talk we explain some properties of the skew RSK dynamics (also another variant of it) and its connections to KPZ models [2].
[1] T. Imamura, M. Mucciconi, T. Sasamoto, Skew RSK dynamics: Greene invariants, affine crystals and applications to $q$-Whittaker polynomials, Forum of Mathematics, Pi (2023), e27 1–101 (arXiv: 2106.11922).
[2] T. Imamura, M. Mucciconi, T. Sasamoto, Solvable models in the KPZ class: approach through periodic and free boundary Schur measures, arXiv: 2204.08420
Konstantin Khanin: On KPZ universality and iterates of random monotone maps
In this talk we will discuss a geometrical approach to the problem of the KPZ universality. Instead of looking at the height (interface) function and Airy processes, we will focus on the statistics of shocks and points of concentration of mass. We will also discuss the connection with the problem of the coalescing Brownian motions and coalescing Fractional Brownian motions.
Percy Deift: Asymptotics for polynomials orthogonal with respect to a weight with a logarithmic singularity
Slides Part 1 Slides Part 2 Video
A.Magnus has made a conjecture concerning the asymptotics of the recurrence coefficients for polynomials orthogonal with respect to a weight on an interval [a,b] with a logarithmic singularity at a or at b. The speaker will show how to verify this conjecture in two special cases. The standard Riemann-Hilbert/steepest descent method does not apply directly as a local parametrix, a key tool in the standard method, in the neighborhood of the logarithmic singularity, is not known. This is joint work with Oliver Conway and Mateusz Piorkowski.
Neil O’Connell: Discrete Whittaker processes
We consider a Markov chain on reverse plane partitions (of a given shape) which is closely related to fundamental Whittaker functions and the Toda lattice. This process has non-trivial Markovian projections and a unique entrance law starting from the reverse plane partition with all entries equal to plus infinity. We also discuss connections with imaginary exponential functionals of Brownian motion, a semi-discrete polymer model with purely imaginary disorder, interacting corner growth processes and discrete delta-Bose gas, and hitting probabilities for some low rank examples.
Pierre Le Doussal: Weak noise theory of the O'Connell Yor polymer
I will describe the weak noise theory of the OY polymer. In the large deviation regime the most probable evolution of the partition function obeys a classical non-linear system which is a non-standard discretisation of the non-linear Schrodinger equation with mixed initial-final conditions. We show that this system is integrable and obtain its solution through inverse scattering methods. This allows to obtain the large deviation rate function of the free energy of the polymer through its conserved quantities and study its convergence to the large deviations of the KPZ equation.
Based on: A. Krajenbrink, P. Le Doussal, arXiv:230701172
Guillaume Barraquand: Stationary measures in the open KPZ class
A two-sided Brownian motion on R remains invariant under the KPZ equation dynamics, up to a global height shift. For the KPZ equation on an interval with boundary conditions, however, stationary measures are more complicated, and have been characterized only recently through the analysis of discrete models. They can be described in terms of two Brownian motions reweighted by exponential functionals. I will review these results and explain why the appearance of such measures is very natural from the point of view of line ensembles.
Nikolaos Zygouras: From the intermediate disorder regime to the Critical 2d Stochastic Heat Flow
The intermediate disorder regime was introduced by Alberts-Khanin-Quastel as means of approximating the stochastic heat equation via polymer models. I will give a review of this method and how this provides a fruitful path to also study the effect of disorder in more general statistical mechanics models. Emphasis will be given on the case of the stochastic heat equation at the critical dimension 2, which gives rise to the Critical 2d Stochastic Heat Flow.