Probability and Stochastic Analysis Seminar 

We consider a Brownian particle in a divergence-free drift, where the vector field is a stationary random field exhibiting "critical" correlations. Predictions from physicists in the 80s state that, almost surely, this process should behave like a "sped-up" Brownian motion at large scales, with variance at time~$t$ being of order~$t \sqrt{\log t}$. In joint work with Ahmed Bou-Rabee and Tuomo Kuusi, we give a rigorous proof of this prediction using an iterative quantitative homogenization procedure, which is a way of formalizing a renormalization group argument. We consider the generator of the process and coarse-grain this operator, scale-by-scale, across an infinite number of scales. The random swirls of the vector field at each scale enhance the effective diffusivity. As we zoom out, we obtain an ODE for the effective diffusivity as a function of the scale, to deduce that it diverges at the predicted rate. Meanwhile, new coarse-graining arguments allow us to rigorously (and quenchedly) integrate out the smaller scales and prove the scaling limit.

We investigate the estimation of the interaction function for a class of McKean-Vlasov stochastic differential equations. The estimation is based on observations of the associated particle system at time $T$, considering the scenario where both the time horizon $T$ and the number of particles $N$ tend to infinity. Our proposed method recovers polynomial rates of convergence for the resulting estimator. This is achieved under the assumption of exponentially decaying tails for the interaction function. Additionally, we conduct a thorough analysis of the transform of the associated invariant density as a complex function, providing essential insights for our main results.

This is based on a joint work with Joffrey Mathien. We establish new instances of the cutoff phenomenon for geodesic paths and for the Brownian motion on compact hyperbolic manifolds. We prove that for any fixed compact hyperbolic manifold, the geodesic path started on a spatially localized initial condition exhibits cutoff. Our work also extends results obtained by Golubev and Kamber on hyperbolic surfaces of large volume to any dimension. More generally, we will discuss ongoing works on the cutoff phenomenon in mixing dynamical systems.

I will discuss the phi42 dynamics, a singular stochastic partial differential formally corresponding to the Langevin dynamics for the phi42 field theory model. The goal is to characterise infinite volume invariant measures for these dynamics. The associated phi42 theory is known to undergo phase transitions as parameters in the model are varied, with the field transitioning from short-distance correlations (in the sense that the susceptibility of the measure defined on a finite box is bounded uniformly on the box size) to long-distance correlations. One expects this behaviour to be reflected in the dynamics, with a unique infinite volume invariant measure in absence of phase transition and possibly more than one in the strongly correlated regime. We prove that this picture is indeed correct in the sense that there is a unique invariant measure for the dynamics whenever the susceptibility is finite. This is done by adapting to the field-theory setting the Holley-Stroock-Zegarlinski approach to uniqueness for statistical mechanics models. This approach is based on a volume-independent bound on the log-Sobolev constant of the associated dynamics, together with crude, model-independent bounds. In the phi42 case the log-Sobolev has recently been established, but substantial work is required to adapt the other parts of the argument. The talk is based on joint work in progress with R. Bauerschmidt and H. Weber.

We begin by introducing Markov processes indexed by Lévy trees, a notion which was developed in a series of works by Duquesne, Le Gall and Le Jan, and that has seen multiple applications in recent years. We will then present the main aspects of a theory, developed in two recent works with Armand Riera, that describes the evolution of a Markov process indexed by a Lévy tree between visits to a regular, instantaneous point of the state space. Despite the radically different setting, we will see that our results share strong similarities with the celebrated Itô excursion theory for Brownian motion. An excursion theory for Brownian motion indexed by the Brownian tree was previously developed by Le Gall and Abraham [JEMS;18], and in particular we recover their results by different methods. Our work is motivated by applications in random geometry.

In theoretical population genetics, it is customary to describe gene genealogies by averaging over the pedigree (the global graph of reproductive relationships). This assumption is built into state-of-the-art statistical tools, classical work on the Kingman coalescent, and related work on multiple mergers. However, this tradition of averaging over the pedigree is questionable because there is only one population pedigree, and all genetic information is passed through this same pedigree. Here we study how the pedigree influences the genealogical relationships of a sample of gene copies at a single genetic locus. We perform this study through the lens of two different diploid exchangeable models: a Wright-Fisher model with occasional big families and a Moran model with partial selfing. For each model, we obtain a novel scaling limit for the conditional genealogy of an arbitrary sample of size n, as the population size N tends to infinity. These scaling limits retain essential information of the population pedigree. Our results offer new insight for ancestral inference and understanding of multi-locus data from populations whose single-locus genealogies are multiple merger coalescents. Joint work with F. Alberti, M. Birkner, D. Diamantidis, M. Newman, and J. Wakeley.

While the mathematical foundations of Generalized Hydrodynamics (GHD) are still incomplete, the theory has proven to be a powerful tool for obtaining accurate approximations of correlation functions in various integrable models. Notably, H. Spohn applied GHD to compute the correlation function of the Toda lattice. In this talk, we focus on the Volterra lattice, another integrable system. We introduce its Generalized Gibbs Ensemble (GGE) and establish a connection with the Anti-symmetric β-ensemble, a well-known random matrix model. This link enables us to explicitly determine the density of states for the Volterra lattice in terms of the corresponding quantity in the random matrix model. Using this result, we apply GHD to derive a linear approximation of the correlation function for the Volterra lattice. This talk is based on the following paper: G. M., Generalized Hydrodynamics for the Volterra Lattice: Ballistic and Nonballistic Behavior of Correlation Functions. J. Phys. A: Math. Theor. DOI: 10.1088/1751-8121/ad742b

Planar random growth processes occur widely in the physical world. Examples include diffusion-limited aggregation (DLA) for mineral deposition and the Eden model for biological cell growth. One approach to mathematically modelling such processes is to represent the randomly growing clusters as compositions of conformal mappings. In 1998, Hastings and Levitov proposed one such family of models, which includes versions of the physical processes described above. An intriguing property of their model is a conjectured phase transition between models that converge to growing disks, and 'turbulent' non-disk like models. In previous work with Norris and Silvestri, we have shown that the global fluctuations present in these models exhibit behaviour that can be interpreted as the beginnings of a macroscopic phase transition from disks to non-disks. In this talk I will discuss work in progress with Larissa Richards in which we explore how the correlation structure of local fluctuations near the cluster boundary changes at the point of phase transition.

Functional inequalities provide information on the structure of a probability measure and on the relaxation of associated stochastic dynamics to equilibrium. In this talk, we will describe a multiscale analysis for decomposing high-dimensional measures into simpler structures and derive from it functional inequalities. The strategy is based on the renormalization group method used in statistical physics to study the distribution of interacting particle systems. We will also review other related developments and in particular show that this decomposition of measures can be interpreted in terms of measure transport. The talk is based on joint works with R. Bauerschmidt and Benoit Dagallier.

In this talk, we will present recent studies on the long-term behavior of the solution to the nonlinear stochastic heat equation $\partial u/\partial t - \frac{1}{2}\Delta u = b(u)\dot{W}$, where $b$ is assumed to be a globally Lipschitz continuous function and the noise $\dot{W}$ is a centered, spatially homogeneous Gaussian noise that is white in time. We identify a set of conditions on the initial data, the correlation measure, and the weight function $\rho$, which together guarantee the existence of an invariant measure or a limiting random field in the weighted space $L^2_\rho(\mathbb{R}^d)$. In particular, our results include the parabolic Anderson model (i.e., the case when $b(u) = \lambda u$) starting from unbounded initial data such as the Dirac delta measure. This study has implications for the study of continuous random polymers. This talk is based on joint work with Nicholas Eisenberg and another collaboration with Ouyang Cheng, Samy Tindel, and Panqiu Xia.

we consider particles following a diffusion process in a multi-well potential and attracted by their barycenter. It is well-known that this process exhibits phase transitions: at high temperature, the mean-field limit has a single stationary solution, the N-particle system converges to equilibrium at a rate independent from N and propagation of chaos is uniform in time. At low temperature, there are several stationary solutions for the non-linear PDE, and the limit of the particle system as N and t go to infinity do not commute. We show that, in the presence of multiple stationary solutions, it is still possible to establish local convergence rates for initial conditions starting in some Wasserstein balls (this is a joint work with Julien Reygner). In terms of metastability for the particle system, we also show that for these initial conditions, the exit time of the empirical distribution from some neighborhood of a stationary solution is exponentially large with N and approximately follows an exponential distribution, and that propagation of chaos holds uniformly over times up to this expected exit time (hence, up to times which are exponentially large with N). Spin glasses are models of statistical mechanics in which a large number of elementary units interact with each other in a disordered manner. In the simplest case, there are direct interactions between any two units in the system, and I will start by reviewing some of the key mathematical results in this context. For modelling purposes, it is also desirable to consider models with more structure, such as when the units are split into two groups, and the interactions only go from one group to the other one. I will then discuss some of the technical challenges that arise in this case, as well as recent progress.

A transition matrix U on a tree T is said to be almost upper directed if the allowed steps are from a node to its parent or its descendants. In this talk, as a warm-up, I will start with the case where the tree is N and I will characterise the recurrence, positive recurrence, and invariant distribution of these transition matrices. I will then present the theorems for general trees and a technique that allows one to compute an invariant distribution at a given vertex without requiring knowledge of the full invariant measure. These results encompass the case of birth and death processes (BDPs), which possess almost upper directed transition matrices. Their properties were studied in the 1950s by Karlin and McGregor, whose approach relies on deep connections between the theory of BDPs, the spectral properties of their transition matrices, the moment problem, and the theory of orthogonal polynomials. Our approach is mainly combinatorial and uses elementary algebraic methods. This talk is based on two joint works with J.-F. Marckert.

Spin glasses are models of statistical mechanics in which a large number of elementary units interact with each other in a disordered manner. In the simplest case, there are direct interactions between any two units in the system, and I will start by reviewing some of the key mathematical results in this context. For modelling purposes, it is also desirable to consider models with more structure, such as when the units are split into two groups, and the interactions only go from one group to the other one. I will then discuss some of the technical challenges that arise in this case, as well as recent progress.

Two core objects in the KPZ universality class are the KPZ fixed point (KPZFP) and the directed landscape (DL). They serve as the scaling limits of random growth processes and random planar geometry, respectively. Moreover, the KPZFP can be obtained as marginals of the DL. A central problem in this field is establishing convergence to both objects. This has been achieved for a few models with exactly solvable structures. Beyond exact solvability, only convergence to the KPZFP has been proven for general 1D exclusion processes. In this talk, I will discuss a recent work with Duncan Dauvergne that uncovers a new connection between these objects: the KPZFP uniquely characterizes the DL. Leveraging this fact, we establish convergence to the DL for a range of models, including some without exact solvability: general 1D exclusion processes, various couplings of ASEPs (e.g., the colored ASEP), the Brownian web and random walk web distances, and directed polymers.

We consider a generalization of the classical perimeter, called nonlocal bi-axial discrete perimeter, where not only the external boundary of a polyomino $\mathcal{P}$ contributes to the perimeter, but all internal and external components of $\mathcal{P}$.

Formally, the nonlocal perimeter $Per_{\lambda}(\mathcal{P})$ of the polyomino $\mathcal{P}$ with parameter $\lambda>1$ is defined as:

$$ Per_{\lambda}(\mathcal{P}):=\sum_{x \in \mathbb{Z}^2 \cap \mathcal{P}, \, y \in \mathbb{Z}^2 \cap \mathcal{P}^c} \frac{1}{d^{\lambda}(x,y)} $$

where $d^{\lambda}(x,y)$ is the fractional bi-axial function defined by the relation:

$$ \frac{1}{d^{\lambda}(x,y)} := \frac{1}{|x_2-y_2|^\lambda}\textbf{1}_{\{ x_1=y_1, \, x_2 \neq y_2\}} + \frac{1}{|x_1-y_1|^{\lambda}} \textbf{1}_{\{ x_2=y_2, \, x_1 \neq y_1\}} $$

with $x=(x_1,x_2)$, $y=(y_1,y_2)$ and $\mathcal{P}^c=\mathbb{R}^2 \setminus \mathcal{P}$.

We tackle the nonlocal discrete isoperimetric problem analyzing and characterizing the minimizers within the class of polyominoes with a fixed area $n$.

The solution of this isoperimetric problem provides a foundation for rigorously investigating the metastable behavior of the long-range bi-axial Ising model.

We consider the symmetric inclusion process on a general finite graph. In the log-concave regime, we establish universal upper and lower bounds for the spectral gap of this process in terms of the spectral gap of the single-particle random walk, thereby verifying the celebrated Aldous' conjecture, originally formulated for the interchange process. Next, in the general non-log-concave regime, we prove that the conjecture does not hold by investigating the so-called metastable regime when the diffusivity constant vanishes in the limit. This talk is based on joint works with Federico Sau.

In this presentation we study the limiting distribution for the joint-law of the largest and the smallest singular values for random circulant matrices with generating sequence given by independent and identically distributed random elements satisfying the so-called Lyapunov condition.

Under an appropriated normalization, the joint-law of the extremal singular values converges in distribution, as the matrix dimension tends to infinity, to an independent product of Rayleigh and Gumbel laws.

The latter implies that a normalized condition number converges in distribution to a Fréchet law as the dimension of the matrix increases. Roughly speaking, the condition number measures how much the output value of a linear system can change by a small perturbation in the input argument.

The proof relies on the celebrated Einmahl--Komlós--Major--Tusnády coupling.

This is based in a paper with Paulo Manrique, Extremes 2022.

In this talk, I will present recent results on the contact process on two specific types of scale-free, inhomogeneous random networks that evolve either through edge resampling or by resampling entire neighborhoods of vertices. Depending on the type of graph, the selected stationary dynamic, the tail exponent of the degree distribution, and the updating rate, we identify parameter regimes that result in either fast or slow extinction. In the latter case, we determine metastable exponents that exhibit first-order phase transitions. This is joint work with Emmanuel Jacob (ENS Lyon) and Peter Mörters (Universitat zu Köln).

The stochastic Curie-Weiss model is probably the simplest example of a non-trivial Glauber dynamics. This model has been extensively studied, and in the high-temperature regime, Levin-Luczak-Peres computed the mixing time up to an optimal constant of order O(n). In the classical definition of mixing time, one takes the least-favorable case as initial condition of the dynamics. A natural question to tackle is the dependence of the mixing time on the initial condition of the dynamics. In order to solve this question, we develop a new framework, which we call sharp convergence. We show sharp convergence of the Curie-Weiss model in the whole high-temperature regime, including non-zero magnetic field, and as an application we compute the mixing time of the Curie-Weiss model up to order o(n), and we also show that the mixing time improves if we take initial conditions with the ‘right’ density. Joint work with Freddy Hernández (Universidad Nacional de Colombia)

We consider the classical 2-opinion dynamics known as the voter model on finite graphs. It is well known that this interacting particle system is dual to a system of coalescing random walkers and that under so-called mean-field geometrical assumptions, as the graph size increases, the characterization of the time to reach consensus can be reduced to the study of the first meeting time of two independent random walks starting from equilibrium.

As a consequence, several recent contributions in the literature have been devoted to making this picture precise in certain graph ensembles for which the above mentioned meeting time can be explicitly studied. I will first review this type of results and then focus on the specific geometrical setting of random regular graphs, both static and dynamic (i.e. edges of the graphs are rewired at random over time), where in recent works we study precise first order behaviour of the involved observables. We will in particular show a quasi-stationary-like evolution for the discordant edges (i.e. with different opinions at their end vertices) which clarify what happens before the consensus time scale both in the static and in the dynamic graph setting. Further, in the dynamic geometrical setting we can see how consensus is affected as a function of the graph dynamics.

Based on recent and ongoing joint works with Rangel Baldasso, Rajat Hazra, Frank den Hollander and Matteo Quattropani.