In this talk I will discuss a polynuclear growth model in half space with two external sources. I will present the strategy developed to study the model in the full space setting (Baik-Rains '00), 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 equation; then I will underline the differences with respect to the half space case. This result also proves a conjecture by Barraquand-Krajenbrink-Le Doussal '22 on the distribution of the stationary KPZ equation on the half line. Based on joint work with M. Cafasso, D. Ofner, H. Walsh.

The Stochastic Burgers Equation (SBE) was introduced in the eighties by van Beijren, Kutner and Spohn as a way to encode the fluctuations of driven diffusive systems with one conserved quantity (e.g. ASEP). In the subcritical dimension d = 1, it coincides with the derivative of the KPZ equation whose large-scale behaviour is polynomially superdiffusive and given by the KPZ Fixed Point, and in the super-critical dimensions d > 2, it was recently shown to be diffusive and rescale to a biased Stochastic Heat equation. At the critical dimension d = 2, the SBE was conjectured to be logarithmically superdiffusive with a precise exponent but this has only been shown up to lower order corrections. In the present talk, we pin down the logarithmic superdiffusivity by identifying exactly the large-time 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 the first superdiffisive scaling limit result for a critical SPDE, beyond the weak coupling regime.

In quantum physics, and more specifically in quantum optics, several notions of classical and hence nonclassical state are in use. They rely on the positivity of quasi-probability distributions, specifically the Glauber-Sudarshan or Wigner functions of the state. Characterizing the classical states is in general a difficult task, involving interesting questions of functional and harmonic analyss. In this talk, we will, after reviewing the subject, report on some recent progress.

In this talk we explore the application of central extensions of Lie groups and Lie algebras to derive the viscous quasi-geostrophic (QGS) equations, with and without Rayleigh friction term, on the torus as critical points of a stochastic Lagrangian. We begin by introducing central extensions and proving the integrability of the Roger Lie algebra cocycle $\omega_\alpha$, which is used to model the QGS on the torus. Incorporating stochastic perturbations, we formulate two specific semi-martingales on the central extension and study the stochastic Euler-Poincaré reduction. Specifically, we add stochastic perturbations to the $\mathfrak{g}$ part of the extended Lie algebra $\widehat{\mathfrak{g}} = \mathfrak{g} \rtimes_{\omega_\alpha} \mathbb{R}$ and prove that the resulting critical points of the stochastic right-invariant Lagrangian solve the viscous QGS equation, with and without Rayleigh friction term.

We construct a time-dependent solution to the Smoluchowski coagulation equation with a constant flux of dust particles entering through the boundary at zero. The dust is instantaneously converted into particles and these solutions, that we call flux solutions, have linearly increasing mass. The construction is made for a general class of non-gelling coagulation kernels for which stationary non-equilibrium solutions exist. In the complementary regime, no flux solution is expected to exist. Flux solutions are expected to converge to a stationary solution in the large time limit. We show that this is indeed true in the particular case of the constant kernel with zero initial data. (Based on a joint work with Aleksis Vuoksenmaa - U. Helsinki)

Duality is an important concept in the study of stochastic interacting particle systems. For arbitrary initial measures duality expresses expectations of a family of functions at time $t$ in terms of the transition probability of a dual process which may be simpler to analyse. Focussing on countable state space we discuss duality from the perspective of the generator. Unlike the more traditional approach of looking at duality in a pathwise manner this allows us to understand straightforwardly how dualities arise from symmetries, or more generally, from invariant subspaces of the generator and leads to constructive methods for finding useful dualities. Also the new concept of reverse duality comes out naturally. It yields the full probability measure of the process at time $t$ for a family of initial measures in terms of transition probabilities of the dual process and thus allows for the computation of arbitrary expectation values.

Duality is an important concept in the study of stochastic interacting particle systems. For arbitrary initial measures duality expresses expectations of a family of functions at time $t$ in terms of the transition probability of a dual process which may be simpler to analyse. Focussing on countable state space we discuss duality from the perspective of the generator. Unlike the more traditional approach of looking at duality in a pathwise manner this allows us to understand straightforwardly how dualities arise from symmetries, or more generally, from invariant subspaces of the generator and leads to constructive methods for finding useful dualities. Also the new concept of reverse duality comes out naturally. It yields the full probability measure of the process at time $t$ for a family of initial measures in terms of transition probabilities of the dual process and thus allows for the computation of arbitrary expectation values.

The Kardar-Parisi-Zhang (KPZ) equation is a singular stochastic partial differential equation which describes the random interface growth in a universal way. Indeed, the KPZ equation has been derived from various types of microscopic systems through a scaling procedure, which phenomenon is referred to as the weak KPZ universality. In this talk, I will introduce two typical regimes from which the KPZ equation is derived in the limit: the weakly asymmetric regime, and the strongly asymmetric regime with the high-temperature limit. After showing some recent results in each of these regimes, I will show some conjecture which enables us to obtain a comprehensive description of the weak KPZ universality in interacting particle systems.

In this talk I will show how one can use stochastic duality to close evolution equations for time-dependent correlation functions and how to use these equations to obtain $L^∞$ bounds for these functions through a random walk approach.

I will give some examples of microscopic models for which this technique can be applied and obtain for those the decay of 2-point correlation functions. If time allows, I will explain the main difficulties for the case of $k$-point correlation functions for $k > 2$.

Based on joint works with Chiara Franceschini, Patrícia Gonçalves and Milton Jara.

We consider a purely harmonic chain of oscillators perturbed by a stochastic noise. Under this perturbation, the system exhibits two conserved quantities: volume and energy. At the hydrodynamic level, in the diffusive time scale, we show that depending on the strength of the Hamiltonian dynamics, energy and volume evolve according to either a system of autonomous heat equations or according to a non-linear system of coupled parabolic equations. At the level of fluctuations, we show that, also in diffusive time scale, under any initial measure, the volume fluctuation field converges. The proofs are based on a precise analysis of the two-point correlation function and a uniform fourth moment bound. We also discuss some open problems and the technical issues faced when studying higher order correlation functions in this models of stochastic oscillators. Joint work with Patricia Gonçalves (IST Lisbon) and Kohei Hayashi (University of Osaka).

In this presentation, we study an ordinary differential equation with a unique degenerate attractor at the origin, perturbed by the addition of Brownian noise with a small parameter that regulates its magnitude. Under general conditions, for any fixed noise magnitude, the solution to this SDE converges exponentially fast in total variation distance to its unique equilibrium distribution as time goes by.

We suitably accelerate the random dynamics and establish that this convergence occurs in a sharp form. More precisely, the total variation distance between the accelerated dynamics and its equilibrium distribution converges to a non-degenerate limiting profile. This profile corresponds to the total variation distance between the marginal distribution of an appropriately defined SDE, which comes down from infinity, and its associated equilibrium distribution. We point out that the limiting profile that emerges from this convergence is not a step function, which is typically observed in the context of the cut-off phenomenon for random processes.

This talk is based in a paper in SPA 2025 with Conrado da Costa (Durham University, UK) and Milton Jara (IMPA, Brazil).

In this talk I will review the 1998 Jordan-Kinderlehrer-Otto interpretation of Fokker-Planck equations as a Wasserstein gradient flow in the space of probability measures, based on optimal transport theory. I will also discuss the corresponding derivation of microscopic-to-macroscopic dissipation, in the language of large deviation and short time limits. This provides a somehow canonical recipee to study vatiational structures for diffusion processes.

If time permits I will also discuss a counter-example to this general idea, based on the so-called (reflected) Sticky Brownian Motion SBM.

Based on joint works with JB Casteras, L. Nenna and F. Santambrogio.

We consider a class of random interface models on the one-dimensional discrete torus $T_N$ parametrised by a positive map $\Phi_N$ on $T_N\times\mathbb{R}$. These models share a weak perturbation, namely an asymmetry of order $N^{-\gamma}$ of the direction of growth that switches from up to down and is always towards the direction that reduces the size of the difference of $\Phi_N$ above and below the interface. We specialise to the case of constant $\Phi_N$, so that the asymmetry direction is based on the sign of the area underneath the interface, and study the hydrodynamic limit, stationary correlation functions and equilibrium fluctuations of the interface. We will also discuss the case of a more general $\Phi_N$. Based on joint work with Martin Hairer and Patrícia Gonçalves.

We will discuss the existence of solutions for cubic higher order Schrödinger type equation (NLS) on the whole space with rough initial data. Although such a problem is known to be ill-posed, we show that a randomisation of the initial data yields almost sure local well-posedness. Using estimates in directional spaces, we improve and extend known results for the standard Schrödinger equation in various directions: higher dimensions, more general operators, weaker regularity assumptions on the initial conditions. We will also discuss the existence of invariant measures in the whole space.

In this presentation we study the limiting distribution of the largest singular value, the smallest singular value, and the so-called condition number for random circulant matrices, where the generating sequence consists of independent and identically distributed (i.i.d.) random elements satisfying the Lyapunov condition.

Under an appropriate normalization, the joint distribution of the extremal (minimum and maximum) singular values converges in distribution, as the matrix dimension approaches infinity, to an independent product of Rayleigh and Gumbel laws. As a consequence, the condition number (properly normalized) converges in distribution to a Fréchet law in the large-dimensional limit. Broadly speaking, the condition number quantifies the sensitivity of the output of a linear system to small perturbations in its input. The proof is based on the celebrated Einmahl–Komlós–Major–Tusnády coupling. This work is based on a joint paper with Paulo Manrique (IPN, Mexico), in Extremes 2022.

The Gibbs sampler (a.k.a. Glauber dynamics and heat-bath algorithm) is a popular Markov Chain Monte Carlo algorithm that iteratively samples from the conditional distributions of the probability measure of interest. Under the assumption of log-concavity, for its random scan version, we provide a sharp bound on the speed of convergence in relative entropy. Assuming that evaluating conditionals is cheap compared to evaluating the joint density, our results imply that the number of full evaluations required for the Gibbs sampler to mix grows linearly with the condition number and is independent of the dimension. This contrasts with gradient-based methods such as overdamped Langevin or Hamiltonian Monte Carlo (HMC), whose mixing time typically increases with the dimension. Our techniques also allow us to analyze Metropolis-within-Gibbs schemes, as well as the Hit-and-Run algorithm. This is joint work with Filippo Ascolani and Giacomo Zanella.

To develop a model for non-equilibrium statistical mechanics, the system is typically brought into contact with two thermodynamic reservoirs, known as boundary reservoirs. These reservoirs impose their own particle density at the system's boundary, thereby inducing a current. Over time, a non-equilibrium steady state emerges, characterized by a stationary current value.

Recently, there has been increasing interest in multi-component systems, where various particle species (sometimes referred to as colors) coexist. In such setups, interactions between diferent species are possible alongside the occupation of available sites.

This work focuses on the boundary-driven multi-species stirring process on a one-dimensional lattice. This process extends naturally from the symmetric exclusion process (SEP) when multiple particle species are considered. Its dynamics involve particles exchanging positions with holes or with particles of diferent colors, each occurring at a rate of 1. Additionally, the system interacts with boundary reservoirs that inject, remove, and exchange types of particles.

After defining the process's generator using an appropriate representation of the gl(N) Lie algebra, we establish the existence of an absorbing dual process defined on an extended chain, where the two boundary reservoir are replaced by absorbing extra sites. This dual process shares bulk dynamics with the original but includes extra sites that absorb particles over extended time periods.

This multi-species stirring process can be mapped onto a higher rank open XXX-Heisenberg spin chain, therefore we employ absorbing duality and the matrix product ansatz to derive closed-form expressions for the non-equilibrium steady-state multi-point correlations of the process. This result is reported in [1].

Next, scaling limits of the process are examined, particularly the behavior of the properly scaled empirical density of the process. First, hydrodynamic equations are derived, illustrating typical system behavior (in the spirit of the law of large numbers). Second, fluctuations from this hydrodynamic limit are investigated, revealing a set of Gaussian processes coupled through noise, resembling aspects of the central limit theorem. Finally, large deviation results are reported, describing the probability of rare trajectories deviating from typical behaviors. An additional outcome of this analysis is the identification of a system of hydrodynamic equations featuring a drift due to interaction with an external field. These scaling limit results are reported in [2] and [3].

References

1. F. Casini, R. Frassek, C. Giardinà, Duality for the multispecies stirring process with open boundaries. (2024) J. Phys. A: Math. Theor. 57 295001
2. F. Casini, C. Giardinà, F. Redig, Density Fluctuations for the Multi-Species Stirring Process. (2024) J Theor Probab 37, 33173354.
3. F. Casini, F. Redig, H. van Wiechen, A large deviation principle for the multispecies stirring process. (2024), Arxiv: 2410.20857.

In this work, we deal with the symmetric exclusion process with $k$ slow bonds equally spaced in the torus with $kn$ sites, where the strength of a slow bond is $α n^{-β}$, where $β>1$. For $k$ fixed, it was known (T. Franco, P. Gonçalves, A. Neumann, AIHP’13) that the hydrodynamic limit in the diffusive scaling of this process is given by the heat equation with Neumann boundary conditions, meaning that the system does not allow flux through a slow bond in the limit. In this joint work with Tiecheng Xu and Dirk Erhard, we obtain another three superdiffusive scalings for this system. If $k$ is fixed and the (time) scaling is $n^θ$, where $2< θ<1+β$, the system reaches equilibrium instantaneously at each box between two consecutive slow bonds, being constant in time and space. If $k$ is fixed and $θ=1+β$, the density is spatially constant inside each box, and evolves as the discrete heat equation. And if $θ=1+β$ and $k$ goes to infinity, we recover the continuous heat equation (on the torus).

In a first step, I will present a probabilistic interpretation for a nonlinear, nonlocal PDE on $(0,\infty)$ introduced to model cell polarization. More precisely, the solution of the PDE is the density at time t of a certain diffusion process. This PDE includes a nonlinear boundary attraction term, which translates on the process into a drift term depending on the density of the solution at $x=0$. Then, this nonlinear process can be approximated by a particle system by replacing the value of the density at 0 by an empirical mean of the local times at the edges of N copies of the process. This particle system can be seen as an N-dimensional process living in a non-regular open and with an oblique reflection on the boundary. I will discuss the existence for the particle system, for the nonlinear process and for the PDE and the links between these three objects. In particular, for a sufficiently large attraction coefficient, these objects must necessarily blow-up in finite time.

In a second step I will study the long term behavior of a class of Self-Interacting Diffusions on $\mathbb R$, where the particle is attracted or repelled by its past trajectory, with aging and linear interactions with past positions. The novelty of this model is to introduce a weight depending on past trajectories through a memory kernel to describe the aging phenomenon. In a first step, we study the case where the memory kernel is general of convolution type. We observe three distinct asymptotic regimes when time becomes large depending on the data and the first moment of the integral kernel. We then study a more realistic non-convolution model and prove a similar transition between the first two regimes. At last, we study the particular case of an exponential memory kernel which allows to construct an explicit solution.

Malliavin Calculus, an infinite dimensional calculus on probability spaces, was born with the paper

P. Malliavin. Stochastic calculus of variations and hypoelliptic operators, Proc. Inter. Symp. Stoch. Diff. Eqs. Kyoto 1976, Wiley (1978), 195–263.

and was initially aimed to give a probabilistic counterpart of Hörmander theorem for hypoelliptic operators. Soon it found many developments and other applications within Mathematics (notably in Mathematical Physics), also in Finance.

I will give an introduction to Malliavin Calculus techniques, with a brief reference to some applications.

Malliavin Calculus, an infinite dimensional calculus on probability spaces, was born with the paper

P. Malliavin. Stochastic calculus of variations and hypoelliptic operators, Proc. Inter. Symp. Stoch. Diff. Eqs. Kyoto 1976, Wiley (1978), 195–263.

and was initially aimed to give a probabilistic counterpart of Hörmander theorem for hypoelliptic operators. Soon it found many developments and other applications within Mathematics (notably in Mathematical Physics), also in Finance.

I will give an introduction to Malliavin Calculus techniques, with a brief reference to some applications.

Let $L = (L(t))_{t \geq 0}$ be a multivariate Lévy process with Lévy measure $\nu(dy) = \exp(-f(|y|))dy$ for a smoothly regularly varying function $f$ of index $\alpha > 1$. The process $L$ is renormalized as $X^\varepsilon(t) = \varepsilon L(\tau_\varepsilon t),\ t \in [0,T]$, for a scaling parameter $\tau_\varepsilon = o(\varepsilon^{-1})$, as $\varepsilon \to 0$. We study the behavior of the bridge $Y^{\varepsilon,x}$ of the renormalized process $X^\varepsilon$ conditioned on the event $X^\varepsilon(T) = x$ for a given end point $x \neq 0$ and end time $T > 0$ in the regime of small $\varepsilon$. Our main result is a sample path large deviations principle (LDP) for $Y^{\varepsilon,x}$ with a specific speed function $S(\varepsilon)$ and an entropy-type rate function $I_x$ on the Skorokhod space in the limit $\varepsilon \to 0+$. We show that the asymptotic energy minimizing path of $Y^{\varepsilon,x}$ is the linear parametrization of the straight line between 0 and $x$, while all paths leaving this set are exponentially negligible. We also infer a LDP for the asymptotic number of jumps and establish asymptotic normality of the jump increments of $Y^{\varepsilon,x}$. Since on these short time scales ($\tau_\varepsilon = o(\varepsilon^{-1})$) direct LDP methods cannot be adapted we use an alternative direct approach based on convolution density estimates of the marginals $X^\varepsilon(t),\ t \in [0,T]$, for which we solve a specific nonlinear functional equation.

Stochastic differential equations (SDE) of McKean-Vlasov type or mean-field type are SDE where the solution's law appears inside the equation's coefficients making them more complicated to solve. On the other hand, such equations have been prominently applied in finance, agent dynamics and machine learning. In this mini course we offer an introduction to this framework covering wellposedness of McKean-Vlasov SDE and properties alongside the associated approximating interacting SDE particle systems and corresponding propagation of chaos.

Stochastic differential equations (SDE) of McKean-Vlasov type or mean-field type are SDE where the solution's law appears inside the equation's coefficients making them more complicated to solve. On the other hand, such equations have been prominently applied in finance, agent dynamics and machine learning. In this mini course we offer an introduction to this framework covering wellposedness of McKean-Vlasov SDE and properties alongside the associated approximating interacting SDE particle systems and corresponding propagation of chaos.

The gene regulatory network models all the biochemical reactions between the different species (proteins, mRNA, etc.) present in a cell. A stochastic approach to this network, using jump processes, has been studied during the '70s (Kurtz), and it was established that in large populations, the rescaled process (thought as a concentrations process) converges towards the solution of an EDO, which is entirely deterministic. In order to obtain a random limit, Crudu, Debussche and Radulescu proposed a multiscale model. In this talk, I will propose a relevant stochastic representation of the multiscale model, for which it is possible to obtain the uniform convergence towards a PDMP and a CLT for the fluctuations around this limit.

We investigate the topology and local geometry of various models of random cubical complexes. In the first part, we consider two types of random subcomplexes of the regular cubical grid: percolation clusters and the Eden cell growth model, analyzing their geometric and topological features. In the second part, we study the fundamental group of random 2-dimensional subcomplexes of an n-dimensional cube, identifying interesting threshold phenomena. This latter model serves as a cubical analogue of the Linial–Meshulam model for random simplicial complexes.

Recently, random walks on Hecke algebras were recognized by A. Bufetov as a natural framework for the study of multi-species interacting particle systems. As a corollary, the Mallows measure can be viewed as the universal stationary blocking measure of interacting particle systems arising from random walks on Hecke algebras. Furthermore, the involution in Hecke algebras implies the color-position symmetry, which is a powerful tool for the asymptotic analysis of multi-species interacting particle systems. In this talk, we explore two facets of random walks on Hecke algebras. The first part focuses on the asymptotic behavior of the Mallows measure. In the second part, we consider applications of the color-position symmetry, particularly in the context of shock fluctuations in the half-line open Totally Asymmetric Simple Exclusion Process (TASEP) and Asymmetric Simple Exclusion Process (ASEP).

We first introduce a brief review of the history of Brownian Motion up to the modern experiments where isolated Brownian particles are observed. Later, we introduce a one-space-dimensional wavefunction model of a heavy particle and a collection of light particles that might generate “Brownian-Motion-Like” trajectories as well as diffusive motion (displacement proportional to the square-root of time). This model satisfies two conditions that grant, for the temporal motion of the heavy particle:

1. An oscillating series with properties similar to those of the Ornstein-Uhlenbeck process;
2. A best quadratic fit with an “average” non-positive curvature in a proper time interval.

We note that Planck’s constant and the molecular mass enter into the diffusion coefficient, while they also recently appeared in experimental estimates; to our knowledge, this is the first microscopic derivation in which they contribute directly to the diffusion coefficient. Finally, we discuss whether cat states are present in the thermodynamic ensembles.

(Joint for with W.D. Wick)

File available at https://papers.ssrn.com/sol3/papers.cfm?abstract_id=5287796

I will discuss the asymptotic behavior at initialization of fully connected deep neural networks with Gaussian weights and biases when the widths of the hidden layers go to infinity. The focus of the talk will be on the one-dimensional case and optimal bounds for the total variation distance obtained by means of Stein's method. This is based on a joint work with S. Favaro, B. Hanin, D. Marinucci and G. Peccati.

Given a finite transitive graph and $n$ particles labeled with numbers ${1,2,3,\dots,n}$, we place these particles on the set of vertices at random. Then, we let the particles evolve as a system of coalescing random walks: each particle performs a continuous-time simple random walk (SRW) and whenever two particles meet, they merge into one particle which continues to perform a SRW. At each time t, consider the partition $P_t$ of ${1,2,3,\dots,n}$ induced by the equivalence relation: $i\sim j$ when particles $i$ and $j$ occupy the same vertex at time $t$. We show that the Kingman $n$-coalescent model emerges as a scaling limit for $(P_t)$, as $n$ is fixed and the size of the graph goes to infinity.

Consider the Simple Symmetric Exclusion Process (SSEP) $\eta_t$ on $\mathbb Z^d$. Let $\Gamma_t=\int_0^t \eta_s(0)ds$ be the occupation time of the origin up to time $t$. In this talk I will discuss about the limit of $\frac{\Gamma_{tn^2}}{\beta_{d,n}}$ for some properly chosen $\beta_{d,n}$. I will first review results in the equilibrium setting, then report some progresses made in the non-equilibrium setting.

In this talk I will present the status on quantum communications in Portugal. From the Laboratory to the operational network what has been done through the main actors in industry, academia and public institutions in Portugal. Starting with the development of national technology under a European Defence project, DISCRETION, to the deployment of the first EuroQCI (the European Quantum Communication Infrastructure) segment in Portugal, PTQCI. I will show that quantum communications is no longer a science project, but it is on the heart of sovereignty in Europe.

In this talk, we present a generalised second-order Boltzmann-Gibbs principle for conservative interacting particle systems on a lattice whose stationary measures are not of product type and not invariant under particle jumps. The result, which requires neither a spectral gap bound nor an equivalence of ensembles, extends the classical framework to settings with correlated invariant measures and is based on quantitative bounds for the correlation decay. As an application, we show that the equilibrium density fluctuations of the Katz-Lebowitz-Spohn model, for a suitable choice of parameters, converge under diffusive scaling to the stationary energy solution of the stochastic Burgers equation. Based on joint work with P. Gonçalves and G. M. Schütz.

This talk investigates the mathematical structure of path measures, both from a measure-theoretical perspective and through stochastic differential equations. The realization of path measures as Langevin systems hinges on the pivotal role of second-order Hamilton-Jacobi-Bellman equations, which form the foundation of stochastic geometric mechanics and applications in stochastic thermodynamics. We explore the emergence of the Onsager-Machlup functional in large deviation theory, the rates of entropy production in irreversible thermodynamic processes, and entropy minimization problems encoded in stochastic geometric mechanics.

We use the framework of Quantitative Hydrodynamics to derive a CLT around its hydrodynamic limit for an interacting particle system inspired by integrate-and-fire neuron models. The hydrodynamic limit of this model was originally derived by Chariker, De Masi, Lebowitz and Presutti, and as an important intermediate step we show that this convergence holds at optimal $L^2$-speed. Joint with Milton Jara and Yangrui Xiang.

To a forward-backward stochastic differential equation is naturally associated a quasilinear partial differential equation. In the spirit of the work of Gaeta, we develop a theory of symmetries for FBSDE's. These symmetries turn out to be related to the symmetries (in Olver's sense) of the corresponding quasilinear PDE. An analogous theory is developed for BSDE's.

This is joint work with Dr. Anas Ouknine (Rouen).

This talk explores neuromorphic computing, an emerging field at the intersection of computational neuroscience and hardware design, focusing on the potential of using stochastic relaxation in recurrent spiking neural networks to efficiently solve combinatorial optimization problems on state-of-the-art neuromorphic chips. We explore the connection between spiking neural networks and the Ising Model, detailing how optimization problems can be mapped to the energy landscape's ground states and how annealing can be used for stochastic relaxation. Integrating these components introduces challenges, particularly the misalignment of neuronal dynamics and ineffective noise generation. We propose to address them by resorting to chaotic networks and a more bio-plausible neural sampling approach.

In this talk, we will discuss how Yau's relative entropy method can be used to establish a quantitative law of large numbers for the particle density in systems with Glauber dynamics, as well as to determine the mixing times of these processes. The presentation will be illustrated with examples drawn from exclusion processes with reservoirs and reaction–diffusion models.