Plenary Lecture or Open Public Lecture
Transport in Disordered Media
Alessandra Faggionato
Random resistor networks are widely used in Physics to analyze transport in disordered systems, such as mixtures of conducting and non-conducting materials, doped semiconductors, and disordered superconductors. For a broad class of random geometries large-scale conduction can be described using deterministic coefficients derived from stochastic homogenization. Within this universal description, a material behaves as a conductor or an insulator depending on the statistical properties of macroscopic crossings in the resistor network. We will describe sufficient conditions for the emergence of macroscopic conduction. As an application, we will discuss Mott variable-range hopping in doped semiconductors in the low temperature regime which amplifies the effects of disorder, giving rise to the physics Mott’s law.
Time permitting, we will also provide insights into universal laws for AC conduction via random electrical circuits and into the analysis of transport through stochastic interacting particle systems in random environments - moving beyond the mean-field approximation at the basis of random resistor networks.
Keywords: percolation theory, random graphs, stochastic homogenization, interacting particle systems.
On rates in the central limit theorem for a class of convex costs
Florence Merlevède
In this talk, we will shall give estimates not only of the usual quadratic transportation cost, but also of a broader class of convex costs between normalized partial sums associated with real-valued random variables and their limiting Gaussian distribution. For the quadratic transport cost, estimates will be given in terms of weak-dependent coefficients that are well suited to a large class of dependent sequences. This class includes irreducible Markov chains, dynamical systems generated by intermittent maps or strong mixing sequences. We will also present very recent results in the independent framework for a broader class of convex costs that includes quadratic cost as a special case but also certain logarithmic costs such as \(|x| \ln (1+ |x|)\). This talk is based on joint works with J. Dedecker and E. Rio.
Superdiffusive Central Limit Theorem for the critical Stochastic Burgers Equation
Giuseppe Cannizzaro
The Stochastic Burgers Equation (SBE) is a singular, non-linear Stochastic Partial Differential Equation (SPDE) which was introduced in the eighties by van Beijren, Kutner and Spohn to describe, on mesoscopic scales, the fluctuations of stochastic driven diffusive systems with one conserved scalar quantity. The classical example of such systems is the Asymmetric (Simple) Exclusion process (ASEP), an interacting particle system whose large-scale behaviour has fascinated probabilists and mathematicians for more than half a century. In the subcritical spatial dimension d=1, the SBE coincides with the derivative of the celebrated Kardar-Parisi-Zhang equation, which is polynomially superdiffusive and whose fluctuations are described by the KPZ Fixed Point, while in the super-critical dimensions d>2, it was recently shown to be diffusive and rescale to a biased Stochastic Heat equation. The present talk focuses on the critical dimension d=2, which falls outside of the ground-breaking theory of Regularity Structures and whose large-scale behaviour had long been open. In their seminal work, van Beijren, Kutner and Spohn conjecture that the SBE should be logarithmically superdiffusive with a precise exponent but this has only been shown up to lower order corrections (both for ASEP in a landmark paper of H.-T. Yau, and, more recently for the SPDE). We pin down the logarithmic superdiffusivity exactly by identifying the asymptotic behaviour of the so-called diffusion matrix and show that, once the logarithmic corrections to the scaling are taken into account, the solution of the SBE satisfies a central limit theorem. This is joint work with Q. Moulard and F. Toninelli.
The cutoff phenomenon for Markov chains
Justin Salez
The cutoff phenomenon is an abrupt transition from out of equilibrium to equilibrium undergone by certain Markov processes in the limit where the size of the state space tends to infinity. Discovered four decades ago in the context of card shuffling, this surprising phenomenon has since then been observed in a variety of models, from random walks on groups or complex networks to interacting particle systems. It is now believed to be universal among fast-mixing high-dimensional processes. Yet, current proofs are heavily model-dependent, and identifying the general conditions that trigger a cutoff remains one of the biggest challenges in the quantitative analysis of finite Markov chains. In this talk, I will provide a self-contained introduction to this fascinating question, and then describe a recent partial answer based on entropy and curvature. Joint work with Francesco Pedrotti.
What AI will not tell you about white noise
Krzysztof Burdzy
This lecture is a brave attempt to show that human intelligence might still be useful in the era of artificial intelligence. White noise and Brownian motion are closely related fundamental concepts in science and mathematics. Their attraction stems in part from their paradoxical mixture of simplicity and complexity.
The lecture will be illustrated and accessible to a wide audience.
Exchangeability in Continuum Random Trees
Minmin Wang
De Fenetti’s Theorem states that all \(\mathbb N\)-indexed exchangeable sequences of real-valued random variables are mixings of i.i.d sequences. For real-valued random processes with exchangeable increments on \([0, 1]\), Kallenberg’s result \([1]\) provides a complete characterisation of these processes via another mixing relationship.
Continuum random trees are random tree-like metric spaces that arise naturally as scaling limits of various models of discrete random trees. In this talk, we will focus in particular on two subclasses of continuum random trees: the so-called stable trees and inhomogeneous continuum random trees. An analogue of Kallenberg’s Theorem for continuum random trees first appeared as a claim in a paper \([2]\) by Aldous, Miermont and Pitman. They suggested that, in much the same way that a stable bridge process on \([0, 1]\) is a mixing of certain extremal exchangeable processes, stable trees are mixings of inhomogeneous continuum random trees.
We present an outline of a rigorous argument supporting this claim, based on a novel construction that applies to both classes of trees. We will also briefly discuss some implications of this result on critical random graphs.
The talk is based on the paper \([3]\).
Bibliography
\([1]\) O. Kallenberg. “Canonical representations and convergence criteria for processes with interchangeable increments.” Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, vol. 27, 1973, pp. 23–36.
\([2]\) D. Aldous, G. Miermont and J. Pitman. “The exploration process of inhomogeneous continuum random trees, and an extension of Jeulin’s local time identity." Probab. Theory Related Fields, vol. 129, no. 2, 2004, pp. 182–218.
\([3]\) M. Wang. “Stables trees as mixings of inhomogeneous continuum random trees.” Stochastic Process. Appl., vol. 175, 2024.
On estimating Fréchet means
Roberto Imbuzeiro Oliveira
A Fréchet mean (or barycenter) of a distribution \(P\) over a metric space \((\mathcal{X},d)\) is any point \[\mu\in {\rm arg}\min_{m\in\mathcal{X}}\mathbb{E}_{X\sim P}\,d^2(X,m).\] This definition makes sense for any \((\mathcal{X},d)\), and coincides with the expectation of \(P\) if \(\mathcal{X}\) is Euclidean space. This talk considers the problem of estimating Fréchet means from an i.i.d. sample from \(P\), possibly contaminated by adversarial noise. The goal is to obtain optimal or-nearly optimal high probability bounds under weak moment assumptions on \(P\).
Two main settings are considered. The first one consists of Alexandrov spaces where geodesics are “bi-extendible” in some precise quantitative sense, which is rich enough to include the space of probability measures over \(\mathbb{R}^d\) with the Wasserstein-\(2\) distance. Our main theorem for this case refines an analysis of \([1]\), and obtains bounds with the “right” variance parameters The second setting we consider is that of uniformly convex Banach spaces of power type \(2\), where our results seem to be completely new. Both results are obtained via an analysis of certain “trimmed empirical processes” that is based on previous work with Lucas Resende \([2]\).
Bibliography
\([1]\) T. Le Goı̈c, Q. Paris, P. Rigollet and A. Stromme. “Fast
convergence of empirical barycenters in Alexandrov spaces and the
Wasserstein space.” Journal of the European Mathematical Society (2022),
Vol 25(6), 2229–2250.
\([2]\) R. I. Oliveira and L. Resende. “Trimmed sample means for robust
uniform mean estimation and regression.” arXiv:2302.06710.
Minimal surfaces in a random environment:
Ron Peled
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" or "Brownian” 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), corresponding to the well-studied case of (non-integrable) first-passage percolation. Based on joint works with Barbara Dembin, Dor Elboim and Daniel Hadas, with Michal Bassan and Shoni Gilboa and with Michal Bassan and Paul Dario.
On the derivation of mean-curvature flow and its fluctuations from microscopic interactions
Sunder Sethuraman
The emergence of mean-curvature flow of an interface between different phases or populations is a phenomenon of long standing interest in statistical physics. In this talk, we review recent progress with respect to a class of reaction-diffusion stochastic particle systems on an \(n\)-dimensional lattice. In such a process, particles can move across sites as well as be created/annihilated according to diffusion and reaction rates. These rates will be chosen so that there are two preferred particle mass density levels \(a_1\), \(a_2\). .2cm
In the evolution, one may understand, when the diffusion and reaction schemes are appropriately scaled, that a rough interface forms between the regions where the mass density is close to \(a_1\) or \(a_2\). Via notions in the theory of hydrodynamic limits, we discuss when the scaled limit of the particle mass density field in \(n\geq 2\) is a sharp interface flow by mean-curvature. We also discuss the fluctuation field limit of the mass near the forming interface, which informs on the approach to the continuum view in a certain stationary regime in \(n=1,2\).
Critical long-range percolation
Tom Hutchcroft
Many statistical mechanics models on the lattice (including percolation, self-avoiding walk, the Ising model and so on) have natural "long-range" versions in which vertices interact not only with their neighbours, but with all other vertices in a way that decays with the distance. When this decay is described by a power-law, it can lead to new kinds of critical phenomena that are not present in the short-range models. I will describe a new approach to the study of these models that allows us to go well beyond what is known for short-range models, particularly in intermediate dimensions.
Bootstrap percolation and kinetically constrained models: universality results
Toninelli Cristina
Recent years have witnessed significant progress in the study of bootstrap percolation (BP) models. In the initial configuration sites are occupied with probability p. The evolution of BP proceeds in discrete time: empty sites remain empty, while occupied sites become empty if and only if a certain model-dependent neighborhood is already empty. On Z^d there is now a fairly complete understanding of the dynamics starting from random initial conditions, along with a clear universality picture for their critical behavior. Much less is known about their non-monotone stochastic counterpart, namely kinetically constrained models (KCM). In these models each vertex is either infected or healthy and, iff it is infectable according to the BP rules, its state is resampled (independently) at rate one and becomes infected with probability p, and healthy with probability 1-p. These models, introduced and intensively studied in physics literature as toy models of the liquid/glass transition, present both challenging and fascinating mathematical problems. Indeed, the presence of constraints induce non-attractiveness, multiple invariant measures, and the breakdown of many powerful tools (such as coercive inequalities, coupling arguments, and censoring techniques) typically used to study convergence to equilibrium. In this talk, I will present a series of results that establish the full universality picture of KCM in two dimensions. We will see that, compared to those of BP, the universality classes for the stochastic dynamics are richer and the critical time scales diverge more rapidly due to the dominant role of energy barriers. The seminar is based on joint works with I.Hartarsky, L.Marêché, F.Martinelli, and R.Morris.