Abstracts

Ivan Kryven (Utrecht): Partial differential equations leading to random graph models

Abstract: In their limiting regime, many random models can be reduced into an effective macroscopic differential equation. A famous example of such dichotomyis the Brownian motion, which can be represented by a Laplace operator in the limit. This example is a special case of the Fayman-Kac formula that connects random walks and linear partial differential equations (PDEs). It turns out that little analogies of this sort are known for non-linear PDEs. In the same time non-linear PDEs may feature peculiar behaviour, for example, their solution may fail to exist outside of a finite interval of time and characterising this time is typically a change. In this talk we will follow the reverse intuition: we will start with an example of a very specific non-linear particular differential and show that the random model that converges to it is the well-known Erd ̋os-Renyi random graph. We will then show that by naturally relaxing the requirements on our PDE, we can arrive with a larger class of random graph models. Interestingly, this analogy can be further exploited to derive existence domains for the non-linear PDEs without explicitly knowing their analytical solution.

Joint work with Jochem Hoogendijk.

Bruno Hideki Fukushima Kimura (Hokkaido): A Theoretical Approach to the Stochastic Cellular Automata and the Digital Annealer's Algorithm

Abstract: Finding a ground state of a given Hamiltonian of an Ising model on a graph G=(V,E) is an important but hard problem. The standard approach for this kind of problem is the application of algorithms that rely on single-spin-flip Markov chain Monte Carlo methods, such as the simulated annealing based on Glauber or Metropolis dynamics. In this work, we investigate new algorithms, the so-called Digital Annealer's algorithm, and some particular kinds of stochastic cellular automata, the SCA and the $\epsilon$-SCA. We prove that if the temperature drops in time n as 1/\log(n), then the Markov chain asymptotically converges to the ground states. We also provide some simulations of these algorithms and show their superior performance compared to the conventional simulated annealing.

Syota Esaki (Fukuoka): SDE representation of eigenvalues, eigenvectors and overlaps of non-Hermitian matrix-valued Brownian motion

Abstract: In this talk, we talk about stochastic processes associated with eigenvalues and eigenvectors of the non-Hermitian matrix-valued Brownian motion.

The non-Hermitian matrix-valued Brownian motion(nHBM) is a dynamical $N \times N$ random matrix model whose entries are given by i.i.d. complex Brownian motions.

The eigenvalue process associated with nHBM are related to the Ginibre ensemble. As matrix-valued Brownian motion, for example, the eigenvalue process for the Hermitian matrix-valued Brownian motion, which assumes the Hermitian symmetry, has been well studied. This process is known to be the solution of the SDE of the Dyson Brownian motion for $\beta = 2$. On the other hand, the eigenvalue process of nHBM is difficult to give the SDE using only themselves, and the SDE is given by combining the eigenvalue process and the eigenvector process. However, in general, eigenvectors are not uniquely determined from the matrix. Hence, it is necessary to pay attention to how the eigenvectors are determined when considering the time evolution of the eigenvectors. In this talk, we consider the time evolution of the overlaps associated with nHBM, and give an expression formula for the time evolution. Overlaps are quantities that express the non-normality of the matrix, and they appear in the quadratic variations of the eigenvalue process. Hence, it is important for the analysis of the eigenvalue and eigenvector processes. In addition, we can see the representation formula does not depend on how the eigenvectors are taken. This is a joint work with Satoshi Yabuoku(National Institute of Technology, Kitakyushu College).

Akira Sakai (Hokkaido): Stability of the critical behavior of the Ising model against quantum perturbation

Abstract: The transverse-field Ising model is a quantum Ising model whose Hamiltonian consists of the standard two-body part (i.e., the sum of -J(x-y)Z_xZ_y, where J is a nonnegative symmetric function on Z^d, and Z_x is the 3rd Pauli matrix acting only on the Hilbert space C^2 at x) and the sum of -qX_x, where X_x is the 1st Pauli matrix at x and q>0 is the strength of the quantum perturbation. Notice that X_x and Z_x do not commute. By the Lie-Trotter formula, this model can be tranlated into an anisotropic classical Ising model on the (d+1)-dimensional space-time. Bjornberg showed in 2013 that, for the nearest-neighbor model in which J(x) equals a positive constant J if x is a neighbor of the origin and 0 otherwise, the space-time two-point function enjoys the infrared bound and, as a result, the susceptibility exhibits the mean-field behavior in a quadrant in the (J,q)-space if d>4 with positive temperature (or d>3 with zero temperature). However, this does not answer the question of whether the critical behavior (e.g., the way the susceptibility diverges as temperature approaches its critical value) is stable against quantum perturbation, or at least near the critical temperature for the classical Ising model (q=0).

In this talk, I will show the lace-expansion approach to this problem. In particular, I will explain potential difficulty in the conventional lace expansion and how to overcome it by an improved version.

Joint work with Yoshinori Kamijima of NCTS in Taiwan.

Sonja Cox (Amsterdam): Affine infinite-dimensional stochastic covariance models

Abstract: Infinite-dimensional stochastic covariance models are used e.g. for forward prices (of currencies or commodities). Such models require the construction of a flexible yet tractable class of stochastic process taking values in the cone of positive operators. In the finite-dimensional (i.e., matrix valued) setting, so-called `affine covariance models' are popular -- examples include the Wishart processes (1991) and the Barndorff-Nielsen and Stelzer model (2007). A full characterization of the class of all affine processes taking values in the cone of positive semi-definite matrices was given by Cuchiero et al. (2011).

We have constructed a class of affine infinite-dimensional stochastic processes taking values in the cone of positive Hilbert-Schmidt operators that allows for state-dependent jumps. Diffusion causes difficulties: we have shown that an infinite-dimensional Wishart process practically only exist if the initial value of the process is of finite rank -- in this case, the process remains of finite rank.As mentioned above, these affine models are popular due to their tractablility: the characteristic funtion is given in terms of (infinite-dimensional) Riccati equations. Nevertheless, simulations pose some challenges, that I will also briefly discuss.

Joint work with Christa Cuchiero, Sven Karbach, and Asma Khedher.

Rajat Hazra (Leiden): Spectrum of inhomogeneous random graphs

Abstract: The talk is on spectrum inhomogeneous random graphs where the edge connection probabilities are independent. We will discuss the adjacency and Laplacian matrix of the graph. The empirical spectral distribution of the matrices after suitable scaling and centering is shown to have a deterministic limit in probability and we will show how the measure changes from the dense to sparse case. We will also see some results on the largest eigenvalues and its corresponding eigenvector.

Daisuke Shiraishi (Kyoto): Random walk on uniform spanning trees

Abstract: The uniform spanning tree (UST) of a finite connected graph is defined by choosing a spanning tree of the graph uniformly at random, which is known to have connections to various areas including loop-erased random walk and electrical networks. The UST is often considered in the same class as various critical statistical physics models since it shares similar properties such as fractal scaling limits with non-trivial scaling exponents and it is one of the few such models for which rigorous results have been proved even for the three-dimensional case, which is typically the most difficult case to study. In this talk, we will consider the geometric and spectral properties of the UST of the $d$-dimensional hypercubic lattice by studying a random walk on it. While the random walk displays mean-field behavior if $d$ is greater than or equal to $4$, different (nontrivial) exponents describe the asymptotic behavior of several quantities such as heat kernel and mean-square displacement of the random walk below four dimensions. Joint work with 1. Omer Angel, David Croydon, Sarai Hernandez-Torres, and 2. Satomi Watanabe.

Evgeny Verbitskiy (Leiden): Random expansions of numbers and applications

Abstract: Ergodic theory is routinely used to study properties of number theoretic expansions of numbers such as decimal or continued fractions expansions. In fact, one can study random expansions of numbers as well. In the first part of the talk, I will discuss the random version of Lochs' theorem, which compares speeds of convergence of various methods of expansions. In the second part of the talk, I will answer the question posed by Y. Jitsumatsu (Kyushu University) about random beta-encoders and their use as random number generators in cryptography.

Makoto Katori (Chuo): Two-dimensional processes associated with the non-Hermitian matrix-valued Brownian motions

Abstract: It is well known that the Hermitian matrix-valued Brownian motion (BM), H(t), is decoupled into the unitary matrix-valued process diagonarlizing H(t) and obtained eigenvalue process. The latter process is identified with the Dyson BM with a special value of parameter and is realized as the non-colliding BM in one dimension. It is regarded as the dynamical extension of GUE studied in random matrix theory (RMT). In the present talk, we discuss the non-Hermitian matrix-valued BM, M(t), whose eigenvalues move on the complex plane and exhibit a dynamics of the Ginibre ensemble studied in RMT. We impose the bi-orthonormality condition between the left- and the right-eigenvectors at each time. This means that the matrix-valued process diagonalizing M(t) is non-unitary. First I will show movies of computer simulations provided by my students, Yuya Tanaka, Saori Morimoto, and Ayana Ezoe, in order to demonstrate the coupling phenomena of the eigenvalue process and the non-unitary matrix-valued process for a variety of initial conditions. Then I will briefly review the recent results on the SDE representation of the processes obtained by Esaki and Yabuoku. In contrast with the Dyson BM having the repulsive interaction between any pair of particles, the eigenvalue process of M(t) is martingale. The cross-variations of the eigenvalues define a matrix-valued process. It exhibits a time-evolution of the overlap matrix whose statistics was first studied by Chalker and Mehlig for the Ginibre ensemble in RMT. We introduce the time-dependent point processes of eigenvalues weighted by the values of overlap-matrix elements, in addition to the usual eigenvalue point-process. Following the electrostatic analogy reported by Burda et al. (2015) in physics literature, possible hydrodynamic descriptions of these processes on the complex plane are shown.

Remco van der Hofstad (Eindhoven): Ising critical values on random graphs

Abstract: The Ising model is one of the simplest statistical mechanics models that displays a phase transition. While invented by Ising and Lenz to model magnetism, for which the Ising model lives on regular lattices, it is now widely used for other real-world applications as a model for cooperative behavior and consensus between people. As such, it is natural to consider the Ising model on complex networks. Since complex networks are modelled using random graphs, this leads us to study the Ising model on random graphs. Real-world networks tend to be highly inhomogeneous, a fact that is most prominently reflected in their degree distributions having heavy tails as described by power laws. In this talk, we discuss recent results on the Ising model on random graphs. Due to the randomness of the graphs on which the Ising model lives, there are different settings for the Ising model on it. The quenched setting describes the Ising model on the random graph as it is, while the annealed setting takes the expectation w.r.t. the randomness of the graph on both sides of the ratio that defines the Boltzmann distribution. These different settings each describe different physical realities, and interestingly, give rise to three different critical values for random graphs with Poissonian degrees. This talk is based on several joint works with Hao Can, Sander Dommers, Cristian Giardina, Claudio Giberti and Maria Luisa Prioriello.

Frank den Hollander (Leiden): Switching interacting particle systems

Abstract: In this talk we consider three classes of interacting particle systems on the integers: dependent random walks, the exclusion process, and the inclusion process. Particles can switch their jump rate between a fast rate and a slow rate. The limit equations for the macroscopic densities of the fast and the slow particles is the well-studied double diffusivity model, a system of reaction-diffusion equations. In order to investigate the microscopic out-of-equilibrium properties, we analyse a system consisting of two finite layers, for the fast and the slow particles, and add boundary reservoirs in both layers. Inside each layer particles move as before, but particles are injected at the left and absorbed at the right at prescribed rates that depend on the layer. We compute the steady-state distribution and the steady-state current. It turns out that uphill diffusion is possible, i.e., the total flow can be opposite to the gradient imposed by the total injection rate and the total absorption rate. This phenomenon, which cannot occur in a single-layer system, is a violation of Fick's law made possible by the switching between the layers.

Joint work with Simone Floreani, Cristian Giardina, Shubhamoy Nandan and Frank Redig.

Naoki Kubota (Nihon): Lipschitz-type estimates for the frog model with the Bernoulli initial configuration

Abstract: We consider the frog model on the multidimensional lattice, whose initial configuration is according to a Bernoulli distribution. This is an interacting particle system consisting of two types of particles: active and sleeping. Active particles randomly move around on the lattice. On the other hand, although sleeping particles do not move at first, they become active and can move around when touched by active particles. Initially, only the origin has one active particle, and the other sites have sleeping particles according to the Bernoulli distribution with some parameter. Then, the genetic active particle gradually generates active particles by touching sleeping ones, and they propagate across the lattice, with time.

The main object of interest is the so-called time constant, which describes the speed of the spread of active particles. The time constant depends on the parameter of the Bernoulli distribution. Hence, in this talk, we discuss how the change in the parameter of the Bernoulli distribution affects the time constant. More precisely, we provide some Lipschitz-type estimates for the time constant.

Takashi Kumagai (Waseda): Periodic homogenization of non-symmetric discontinuous Markov processes

Abstract: Homogenization problem is one of the classical problems in analysis and probability that is very actively studied until recently. In this talk, we consider homogenization problem for non-symmetric discontinuous Markov processes, in particular L¥'evy-type processes with drifts in periodic media. Under a proper scaling, we show the scaled processes converge weakly to L¥'evy processes on ${¥mathbb R}^d$. We completely characterize the limiting processes when the coefficient function of the drift part is bounded continuous, and the decay rate of the jumping measure is comparable to $r^{-1-¥alpha}$ for $r>1$ in the spherical coordinate with $¥alpha ¥in (0,¥infty)$. Different scaling limits appear depending on the values of $¥alpha$. If time permits, we will briefly discuss our on-going work on quantitative homogenization for L¥'evy-type processes in time-dependent periodic media.

Joint work with with Xin Chen, Zhen-Qing Chen and Jian Wang.

Michel Mandjes (Amsterdam): General Multivariate Hawkes Processes and Induced Population Processes: exact results and large deviations

Abstract: Among multivariate point processes, the class of Hawkes processes, or mutually exciting processes, provides a natural contender for modeling contagion phenomena. In this talk I'll discuss multivariate population processes in which general, not necessarily Markovian, multivariate Hawkes processes dictate the stochastic arrivals.

The first class of results concerns the identification of the time-dependent joint probability distribution, allowing for general intensity decay functions, general intensity jumps, and general sojourn times, in terms of a fixed-point representation.

The second class of results focuses on risk processes driven by multivariate Hawkes arrivals. The main results are (i) a large deviations principle for the cumulative claim process in the light-tailed regime, (ii) the identification of the decay rate of the ruin probability, and (iii) an importance sampling based efficient simulation procedure to estimate rare events.

Makoto Nakashima (Nagoya): Feynman-Kac formula for Schr\"odinger equation with one point interaction

Abstract: It is known that all self-adjoint extensions of Schr\"odinger operator $-\Delta|_{C_0^\infty(\mathbb{R}^d\backslash \{0\})}$ in $L^2(\mathbb{R}^d)$ are given by a one-parameter family $-\Delta_\alpha$ ($\alpha\in (-\infty,\infty]$) in $d=2,3$. In particular, $-\Delta_\alpha$ ($|\alpha|<\infty$) describes a point interaction.

In this talk, we focus on a heat equation $\frac{\partial _t}{\partial t}u=\frac{1}{2}\Delta_\alpha u $ and construct a Feynman-Kac formula to $u^\alpha$ by introducing a new process $W$ and a function $G$.

Yuki Tokushige (Kyushu): Scaling limits of SRWs on the Long-Range Percolation cluster

Abstract: In this talk, we will discuss a Long-Range Percolation (LRP) on $\mathbb{Z}^d$, which is a variant of the classical nearest-neighbor Bernoulli percolation. In particular, we consider a problem concerning the fluctuation of a SRW on the infinite cluster of a LRP. In the paper published in 2013, Crawford and Sly proved that a SRW on the infinite cluster scales to $\alpha$-stable process. However, they completed the proof only when $\alpha\in(0,1)$. It is because the remaining case involves a technical problem to do with a control of short jumps of a SRW, which is related to the fact that sample paths of $\alpha$-stable process have infinite variations for $\alpha\in[1,2)$.

In this talk, we will give a brief overview of this subject and explain our idea to overcome the technical issue depicted above and complete the proof of the case $\alpha\in[1,2)$. This talk is based on a joint work with Noam Berger (Technical Univeraity of Munich).

Ryoki Fukushima (Tsukuba): Distribution of the random walk conditioned on survival among quenched Bernoulli obstacles

Abstract: Consider a simple symmetric random walk that is conditioned to stay on a supercritical percolation cluster up to a large time n. Following a series of works of Sznitman in 1990s, it has recently been shown by Ding and Xu that with high probability, the random walk will be localized in a ball of volume proportional to log n. In this talk, I present the further refinements: (1) this ball is free of obstacles, (2) the limiting one-time distributions of the random walk are obtained. This talk is based on a joint work with Jian Ding, Rongfeng Sun and Changji Xu.

Luca Avena (Leiden): Evolution of discordances in voter dynamics on random regular graphs

Abstract: We consider the classical continuous-time interacting particle system for 2-opinion dynamics known as the voter model on a random d-regular graph. It is well known that this particle system is dual to a system of coalescing random walkers and that in this geometrical setting, as the graph size increases, the time to reach consensus grows linearly with the number of nodes n. We study the time-evolution of the density of the discordant edges (i.e. edges with different opinions at their end vertices) until the consensus is reached. This observable can be thought as the dynamic “perimeter” associated to the density of opinions of one type in the given network (the latter to be seen as the “volume”). While the volume of one opinion evolves as a martingale until the consensus time, the density of discordant edges (i.e. the perimeter) undergoes a very different quasi-stationary-like evolution which we make precise. In particular, starting from a Bernoulli independent assignment of the two initial opinions of parameter u in (1/2,1), we prove that, on time-scale of order one, the fraction of discordant edges goes down and stabilizes to an explicit constant function of d and u related to the meeting time of two random walks in an infinity tree. Then, at time scale n, this density of discordances moves out of the constant plateau and converges in an exponential fashion to zero. Underlying proofs are built on delicate partially new coupling constructions exploiting the locally tree-like geometry, the dual coalescing system and the so-called first visit time lemma.

Joint work with Rangel Baldasso, Rajat Hazra, Frank den Hollander andMatteo Quattropani.

Patrick van Meurs (Kanazawa): Scaling limits of a nonlocally interacting particle system on the discrete torus

Abstract: We consider n particles on the discrete Torus with spacing d. The dynamics are given by simple Kawasaki dynamics and a drift. The drift is given by the interaction force exerted by all other particles. Our interest is not just in passing to the limit as both n tends to infinity and d tends to 0, but also in deriving convergence rates for this convergence. For the limit n to infinity we obtain such a convergence rate by proving propagation of chaos, and for the limit d to 0 we rely on tools from numerical analysis to establish a bound on the difference between the laws of the related processes.

This is joint work with T. Hudson (University of Warwick) and M. A. Peletier (Eindhoven University of Technology).

Yosuke Kawamoto (Okayama): The tail preserving property of interacting Brownian motions in infinite dimentions (canceled)

Abstract: For a random point field with infinitely many particles, we consider interacting Brownian motions which leave this measure invariant. These dynamics have been constructed for many random point fields including long-range interacting systems. Furthermore, several general results have been established to show detailed properties of the processes. However, these frameworks essentially require the tail triviality of random point fields, and this point is a bottleneck because the triviality has been proved only for restricted random point fields. To overcome this, we introduce the tail preserving property in the sense that invariant sets of interacting Brownian motions are elements of the tail $\sigma$-field. In this talk, we shall show that the tail preserving property removes the tail triviality from the assumptions of the frameworks and that this property holds for a quite wide class of infinite particle systems.

Aernout van Enter (Groningen): Symmetric Versus Asymmetric Regularity Properties. Ising Examples, The Roles Of Entropic Repulsion.

Abstract: I discuss the possible (in)equivalence between the asymmetric g-measure property, and the symmetric property of being a Gibbs measure. It turns out that neither property implies the other one, and that in both directions counterexamples exist, of measures constructed by considering Ising models, which have different entropic repulsion behaviours.

Joint work with R. Bissacot, E.O. Endo and A. Le Ny, and with S.B. Shlosman.

Tomohiro Sasamoto (Titech): Mapping macroscopic fluctuation theory for 1D interacting particle systems to classically integrable systems

The large deviation principle for symmetric simple exclusion process(SEP) had been established by Kipnis, Olla, Varadhan in 1989 [1]. A somewhat different formulation, known as the macroscopic fluctuation theory (MFT), was initiated and developed by Jona-Lasinio et al in 2000's [2]. The basic equations of the theory, MFT equations, are coupled nonlinear partial differential equations and have resisted exact analysis except for stationary situation. In this talk we show that a generalization of the Cole-Hopf transformation maps the MFT equations for a class of interacting particle systems including SEP to the classically integrable Ablowitz-Kaup-Newell-Segur(AKNS) system. This allows us to solve the equations exactly in time dependent regime by adapting standard ideas of inverse scattering method.

The talk will be based on collaborations with K. Mallick and H. Moriya [3].

References:

[1] C. Kipnis, S. Olla, S. R. S. Varadhan, Hydrodynamics and large deviations for simple exclusion processes, Comm. Pure Appl. Math., 42:115--137, 1989.

[2] L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, and C. Landim, Macroscopic fluctuation theory, Rev. Mod. Phys., 87:593--636, 2015.

[3] K. Mallick, H. Moriya, T. Sasamoto, Exact solution of the macroscopic fluctuation theory for the symmetric exclusion process, PRL 129, 040601 (2022).