Introduction Room: Centre de conférences Marilyn et James Simons

Mathematical Foundation of Various MCMC Methods

• Akira Sakai (Hokkaido University)

Room: Centre de conférences Marilyn et James Simons Combinatorial optimization problems are ubiquitous in various fields of practical and theoretical interest. The famous traveling salesman problem is one of them. One approach to tackle those problems is to use an Ising model whose Hamiltonian $H$ takes its minimum at a spin configuration, called a ground state, which corresponds to an optimal solution to the corresponding original problem. Standard MCMC methods, such as the Glauber dynamics and the Metropolis algorithm, have been used for decades to sample the Gibbs distribution, which is proportional to $e^{-H/T}$, hence close to the uniform distribution over the ground states when the temperature $T$ is very small. However, those MCMC methods are based on single-spin flip rules, hence prone to being slow. In this talk, I will explain three other MCMC methods, two among which are based on multi-spin flip rules, hence potentially fast. I will show several mathematical results, as well as numerical results to compare which is better in which context. This talk is based on joint work with Bruno Hideki Fukushima-Kimura and many others involved in the CREST project for the past five years.

Refined Cauchy/Littlewood Identities and Their Applications to KPZ Models

• Takashi Imamura (Chiba University)

Room: Centre de conférences Marilyn et James Simons The Cauchy identity is a formula about a sum of a product of two Schur functions over partitions and plays an important role in combinatorics, representation theory, and integrable probability. Some generalizations about such as sums of Macdonald polynomials and skew Shur functions are also known. In this talk, I will report our recent works[1,2] with Matteo Mucciconi (Warwick University) and Tomohiro Sasamoto (Tokyo Institute of Technology) on the identities connecting the sums about the q-Whittaker functions (the case $t=0$ of the Macdonald polynomial) and the skew Schur functions. They can be considered as refinements of the Cauchy/Littlewood identities. We give a proof of them based on algebraic combinatorics: We introduce a deterministic time evolutions called the skew RSK dynamics and show that one can linearize the dynamics by using some techniques of the affine crystal. The combinatorial objects obtained from the linearized one can be seen as building blocks of sum about the q-Whittker functions while those from the skew RSK dynamics itself are associated to the sum about the skew Schur functions. In the language of the integrable probability, the identities can be regarded as relations between two probability measures, the full space/half space q-Whittaker measures and the periodic/free boundary Schur measures. The former measures are related to various KPZ models while the latter ones are typical models of determinantal/Pfaffian and point processes. From these relations we can immediately get the Fredholm determinant/Pfaffian formulas for distribution functions of certain random variables for KPZ models. [1] T. Imamura, M. Mucciconi, and T. Sasamoto, Forum of Mathematics, Pi 11(e27) 1-101 [2] T. Imamura, M. Mucciconi, and T. Sasamoto, arXiv:2204.08420

Fredrickson-Andersen $2$-spin Facilitated Model: Sharp Threshold

• Cristina Toninelli (CNRS & Université Paris Dauphine - PSL)

Room: Centre de conférences Marilyn et James Simons The Fredrickson-Andersen $2$-spin facilitated model (FA-$2$f) on $\mathbb Z^d$ is a paradigmatic interacting particle system with kinetic constraints (KCM) featuring cooperative and glassy dynamics. For FA-$2$f vacancies facilitate motion: a particle can be created/killed on a site only if at least $2$ of its nearest neighbors are empty. We will present sharp results for the first time, $\tau$, at which the origin is emptied for the stationary process when the density of empty sites ($q$) is small. In any dimension $d\geq 2$ it holds $$\tau\sim \exp\left(\frac{d\lambda(d,2)+o(1)}{q^{1/(d-1)}}\right)$$ w.h.p., with $\lambda(d,2)$ the threshold constant for the $2$-neighbour bootstrap percolation on $\mathbb Z^d$. We will explain the dominant relaxation mechanism leading to this result, give a flavour of the proof techniques, and discuss further results that can be obtained via our technique for more general KCM, including full universality results in two dimensions. Joint work with I.Hartarsky and F.Martinelli.

Sharp Interface Limit for Glauber-Kawasaki Process

• Kenkichi Tsunoda (Kyushu University)

Room: Centre de conférences Marilyn et James Simons We discuss scaling limits for Glauber-Kawasaki process. The Glauber-Kawasaki process has been introduced by De Masi, Ferrari and Lebowitz to study a reaction-diffusion equation from a microscopic interacting system. They have derived a reaction-diffusion equation as a limiting equation of the density of particles. This limit is usually called hydrodynamic limit. In this talk, I will focus on several scaling limits related to this hydrodynamic limit. Especially, I will discuss a sharp interface limit for this particle system and its large deviation rate function.

Cutoff for the Transience Time for the SSEP with Traps and the One-Dimensional Facilitated Exclusion Process (FEP)

• Clément Erignoux (INRIA & Université de Lyon 1)

Room: Centre de conférences Marilyn et James Simons The facilitated exclusion process is a toy model for phase separation, where particles can jump to an empty neighboring site iff their other neighboring site is occupied. Because of this kinetic constraint, at low densities $\rho\leq 1/2$, the FEP ultimately reaches a frozen state where particles are all surrounded by empty sites, whereas at large densities $\rho>1/2$, the FEP reaches an ergodic component where it can be mapped to the classical SSEP. In this talk, I will present a new mapping of the FEP to a process that we call SSEP with traps, that displays the same frozen/ergodic phases. I will then focus on the estimation on the transience time needed to reach either an ergodic or frozen state for this model started from the "worst" possible state, which undergoes a cutoff as the size of the system diverges. I will then explore the consequences of this transience time cutoff on the mixing time on the FEP, and on the mixing time of both processes. Based on JW with Brune Massoulié (Université Paris Dauphine).

A Statistical Physics Approach to the Sine Beta Process and Other Random Point Processes

• Mylène Maïda (Université de Lille)

Room: Centre de conférences Marilyn et James Simons The Sine process (corresponding to inverse temperature beta equal to 2) is an ubiquitous determinantal point process. It appears as the bulk limit of many particle systems in various contexts (random matrix ensembles, zeros of L-functions, growth models etc.) Its universality properties are fascinating. There is also a whole family of Sine beta processes, introduced by Valko and Virag, as the bulk limit of Gaussian beta ensembles, for any positive beta. As soon as beta is different from 2, much less is known. I will explain how tools from classical statistical mechanics such as Dobrushin-Lanford-Ruelle (DLR) can be used to better understand their structure. This will be based on a joint work with David Dereudre, Adrien Hardy (Université de Lille), and Thomas Leblé (Université Paris Cité) but I will also review other - more recent - applications.

Dominated Representations, Intersections and Large Deviations

• Ryokichi Tanaka (Kyoto University)

Room: Centre de conférences Marilyn et James Simons We show that mean distortions and growth rates determine rough similarity classes of hyperbolic metrics in groups, and discuss its relation to the rigidity of dominated representations and concentration phenomena for counting measures on large balls. Joint work with Stephen Cantrell (Warwick).

Presentation of the IRL in The University of Tokyo by the Director

• Michael Pevzner (Directeur de French-Japanese Laboratory of Mathematics and its Interactions)

Room: Centre de conférences Marilyn et James Simons

Zeros of Random Power Series with Stationary Gaussian Coefficients

• Tomoyuki Shirai (Kyushu University)

Room: Centre de conférences Marilyn et James Simons The zeros of random power series with i.i.d. complex Gaussian coefficients form the determinantal point process associated with the Bergman kernel. As a natural generalization of this model, we are concerned with zeros of Gaussian power series with coefficients being stationary, centered, complex Gaussian process. The zeros of such analytic Gaussian process have special properties. Our main concern is the expected number of zeros in a disk and we compare it with the i.i.d. coefficients case. When the spectral density of the Gaussian process of coefficients is nice, we discuss the precise asymptotic of the expected number of zeros inside the disk of radius $r$ centered at the origin as $r$ tends to the radius of convergence. Also, we discuss the relationships between the intensity of zeros and spectral density.

Dunkl Operators, Random Matrices and Hurwitz Numbers

• Nizar Demni (Aix-Marseille Université)

Room: Centre de conférences Marilyn et James Simons This talk is concerned with selected probabilistic aspects of Dunkl operators. In the first part, I'll revisit Cepa and Lepingle study of particles on the real line then I'll show how it extends to radial Dunkl processes associated to reduced root sytems. In the second part, I'll talk about the reflected Brownian motion in Weyl chambers. In this respect, I'll exhibit its construction using folding operators and provide the Tanaka-type formula it satisfies. The last part is devoted to the mysterious occurrence of simple Hurwitz numbers in the expression of the Dunkl interwining operator and in particular in the generalized Bessel function (HCIZ integral).

Outliers of Perturbations of Banded Toeplitz Matrices

• Mireille Capitaine (IMT)

Room: Centre de conférences Marilyn et James Simons Let $T_n({\bf a})$ be a $n\times n$ Toeplitz matrix with symbol ${\bf a}\colon \mathbb S^1 \to \mathbb{C}$ given by the Laurent polynomial ${\bf a}(\lambda) = \sum_{k=-r}^s a_k \lambda^k$. We consider the matrix $$ M_n = T_n({\bf a}) + \sigma \frac{X_n}{\sqrt{n}}, $$ where $\sigma >0$ and $X_n$ is some noise matrix whose entries are centered i.i.d. random variables of unit variance. When $n$ goes to infinity, the empirical spectral distribution of $M_n$ converges towards a probability measure $\beta_\sigma$ on $\mathbb{C}$. The objective of this talk is to describe, when $n$ is large, the eigenvalues of $M_n$ in closed regions of $\mathbb{C} \backslash \mbox{support}(\beta_\sigma)$ which we will call the outlier eigenvalues. This is a joint work with Charles Bordenave and Fran\c{c}ois Chapon.

An Exact Solution of the Macroscopic Fluctuation Theory

• Kirone Mallick (CEA & IPhT)

Room: Centre de conférences Marilyn et James Simons The Macroscopic Fluctuation Theory (MFT) is a framework proposed by Bertini, De Sole, Gabrielli, Jona-Lasinio and Landim for diffusive interacting particle systems, in which large deviations functions can be obtained by solving a system of nonlinear PDEs. In this talk, we shall present an exact solution of the MFT for a paradigmatic process, the simple exclusion process, derived by using Inverse Scattering Theory.

Regularity Structures for Quasilinear Singular SPDEs

• Ismael Bailleul (Université de Bretagne Occidentale)

Room: Centre de conférences Marilyn et James Simons I will give an overview of the tools of regularity structures for the study of semilinear and quasilinear parabolic singular stochastic PDEs. No previous knowledge of the domain is needed.

Random Models on Regularity-Integrability Structures

• Masato Hoshino (Osaka University)

Room: Centre de conférences Marilyn et James Simons In the study of singular SPDEs, it has been a challenging problem to obtain a simple proof of a general probabilistic convergence result (BPHZ theorem). Differently from Chandra and Hairer's Feynman diagram approach, Linares, Otto, Tempelmayr, and Tsatsoulis recently proposed an inductive proof based on the spectral gap inequality by using their multiindex language. Inspired by their approach, Hairer and Steele also obtained an inductive proof by using the regularity structure language. In this talk, we introduce an extension of the regularity structure including integrability exponents, and provide a simpler proof of BPHZ theorem. This talk is based on a joint work with Ismael Bailleul (Université de Bretagne Occidentale).

Global Solutions to Quadratic Systems of Stochastic Reaction-Diffusion Equations in Space-Dimension Two

• Julien Vovelle (CNRS & ENS Lyon)

Room: Centre de conférences Marilyn et James Simons With Marta Leocata (LUISS, Roma), we study stochastic perturbations to systems of reaction-diffusion equations, the structure of the noise being deduced from the modeling of chemical reactions in a diffusive regime. I will explain in this talk how to establish the existence of global solutions for four-species quadratic systems in space-dimension two.

Particle Approximation of the Doubly Parabolic Keller-Segel Equation in the Plane

• Milica Tomasevic (CNRS & École polytechnique)

Room: Centre de conférences Marilyn et James Simons In this talk, we study a stochastic system of $N$ particles associated with the parabolic-parabolic Keller-Segel system in the plane. This particle system is singular and non Markovian in that its drift term depends on the past of the particles. When the sensitivity parameter is sufficiently small, we show that this particle system indeed exists for any $N\geq 2$, we show tightness in $N$ of its empirical measure, and that any weak limit point of this empirical measure, as $N\to \infty$, solves some nonlinear martingale problem, which in particular implies that its family of time-marginals solves the parabolic-parabolic Keller-Segel system in some weak sense. The main argument of the proof consists of a "Markovianization" of the interaction kernel: We show that, in some loose sense, the two-by-two path-dependant interaction can be controlled by a two-by-two Coulomb interaction, as in the parabolic-elliptic case. This is a joint work with N. Fournier (Sorbonne Université).

Deviation of Capacity of the Range of Random Walk

• Izumi Okada (Chiba University)

Room: Centre de conférences Marilyn et James Simons We study the capacity of the range of a simple random walk in three and higher dimensions. It is known that the order of the capacity of the random walk range in n dimensions is similar to that of the volume of the random walk range in n-2 dimensions. We find the specific difference between the law of the iterated logarithm and large deviation for the capacity of the random walk range and the volume. This is joint work with Arka Adhikari (Stanford), and Amir Dembo (Stanford).

Scaling Limits of Disordered Systems

• Quentin Berger (Sorbonne Université)

Room: Centre de conférences Marilyn et James Simons I will present some recent results on the scaling limit of disordered systems and some of their consequences. I will mostly focus on the Poland–Scheraga model, also known as the pinning model, which is used to describe DNA denaturation: the question is to know wether (and how) disorder affects its phase transition. I will present some results obtained in a generalized (supposedly more realistic) version of the model, in collaboration with Alexandre Legrand (Université Lyon 1).

Equivalence of Fluctuations Between SHE and KPZ Equation in Weak Disorder Regime

• Shuta Nakajima (Meiji University)

Room: Centre de conférences Marilyn et James Simons The Kardar-Parisi-Zhang (KPZ) equation is a mathematical model that describes the random evolution of interfaces. The equation has become a fundamental model in non-equilibrium statistical physics. Constructing a solution to the KPZ equation in any dimension presents a significant challenge due to its inherent non-linearity. This challenge has resulted in an enduring open problem, particularly in finding solutions in two and higher dimensions. This talk will explore the intriguing connection between the stochastic heat equation (SHE) and the KPZ equation. It offers a rigorous demonstration of the equivalence of fluctuations in these systems in the weak disorder regime for three and higher dimensions. This talk is based on joint work with Stefan Junk (Gakushuin University).

Where Do Random Trees Grow Leaves

• Nicolas Curien (Université Paris-Saclay)

Room: Centre de conférences Marilyn et James Simons Luczak and Winkler (refined by Caraceni and Stauffer) showed that is it possible to create a chain of random binary trees $(T_n : n \geq 1)$ so that $T_{n}$ is uniformly distributed over the set of all binary trees with $n$ leaves and such that $T_{n+1}$ is obtained from $T_{n}$ by adding "on leaf". We show that the location where this leaf must be added is far from being uniformly distributed on $T_n$ but is concentrated on a "fractal" subset of $n^{3(2- \sqrt{3})+o(1)}$ leaves. The full multifractal spectrum of the measure in the continuous setting is computed. Joint work with Alessandra Caraceni and Robin Stephenson.

Geometric Laplacians on Self-Conformal Fractal Curves in the Plane

• Naotaka Kajino (Kyoto University)

Room: Centre de conférences Marilyn et James Simons This talk will present the speaker's ongoing work on “geometrically canonical” Laplacians on self-conformal fractal curves in the plane. The main result is that on a given such curve one can construct a family of Laplacians whose heat kernels and eigenvalue asymptotics “respect” the fractal nature of the Euclidean geometry of the curve in certain nice ways. The idea of the construction of such Laplacians originated from the speaker's preceding studies on the case of a circle packing fractal, i.e., a fractal subset of $\mathbb{C}$ whose Lebesgue area is zero and whose complement in the Riemann sphere $\widehat{\mathbb{C}}:=\mathbb{C}\cup\{\infty\}$ is the union of disjoint open disks in $\widehat{\mathbb{C}}$. He has observed that, on a given such fractal, one can explicitly define a Dirichlet form (a quadratic energy functional) by a certain weighted sum of the standard one-dimensional Dirichlet form on each of the circles constituting the fractal, and that this Dirichlet form “respect” the Euclidean geometry of the fractal in the sense that the inclusion map of the fractal into $\mathbb{C}$ is harmonic with respect to this form. The speaker has also proved that such a Dirichlet form is unique for the classical Apollonian gaskets and that, for some concrete families of self-conformal circle packing fractals including the Apollonian gaskets, the associated Laplacian satisfies Weyl's eigenvalue asymptotics involving the Euclidean Hausdorff dimension and measure of the fractal. It would be desirable if one could extend such results to self-conformal fractals which are not circle packing ones, and the talk will present an extension to the simplest case of self-conformal fractal curves in the plane. The key point of the construction of Laplacians is to use (suitable versions of) the harmonic measure in defining the Dirichlet form BUT to use fractional-order Besov seminorms (with respect to the harmonic measure) of the inclusion map into $\mathbb{C}$ in defining the $L^{2}$-inner product for functions on the fractal.