2023-24

In this talk, we will present some recent progress on using optimal transport theory and PDE methods to analyze the empirical large-time behavior of stochastic processes. Specifically, we studied the asymptotic properties of the occupation measure for Brownian motion on flat tori, by deriving an upper bound that we conjecture to be sharp, in terms of Brownian interlacements on Euclidean space. Our techniques show the potential for generalized application to other types of diffusion processes on weighted manifolds, and a similar asymptotic analysis may be possible for a variety of stochastic models arising in physics, biology, and beyond. Based on a joint work with M. Mariani (arXiv:2307.10325).

We prove that the fluctuations of the eigenvalues converge to the Gaussian Free Field (GFF) on the unit disk. These fluctuations appear on a non-natural scale, due to strong correlations between the eigenvalues. We also study the space--time correlations of those eigenvalues.

Then, motivated by the long-time behaviour of the ODE \dot{u}=Xu, we give a precise estimate on the eigenvalue with the largest real part and on the spectral radius of X, and prove the universality of their Gumbel fluctuations.

The Averaging process (a.k.a. repeated averages) is a mass redistribution model over the vertex set of a graph. Given a graph G, the process starts with a non-negative mass associated to each vertex. The edges of G are equipped with Poissonian clocks: when an edge rings, the masses at the two extremes of the edge are equally redistributed on these two vertices. Clearly, as time grows to infinity, the state of the system will converge (in some sense) to a flat configuration in which all the vertices have the same mass. This very simple process has been introduced to the probabilistic community by Aldous and Lanoue in 2012. However, up to few years ago, there was no graph for which sharp quantitative results on the time needed to reach equilibrium were available. Indeed, the analysis of this process requires different tools compared to the classical Markov chain framework, and even in the case of seemingly straightforward geometries—such as the complete graph or the 1-d torus—it can be handled only by means of non trivial probabilistic and functional analytic techniques. During the talk, I’ll try to give a broad overview of the problem and of its difficulties, and I'll present the few examples that have been completely settled. 

Based on joint work with P. Caputo (Roma Tre) and F. Sau (Università di Trieste)

Abstract:  We formulate the central limit theorem for the winding number and free energy for a directed polymer on 1 + 1 dimensional cylinder. We give formulas for the asymptotic variance in each case and discuss the relation between them. This is a joint work with A. Dunlap (Courant Institute) and Y. Gu (Univ. of Maryland). 

[1] Y. Gu, T. Komorowski„ KPZ on torus: Gaussian fluctuation, to appear in Ann. Inst. H. Poincare, Prob. and Stat., https://arxiv.org/abs/2104.13540

[2] Y. Gu, T. Komorowski„ Fluctuations of the winding number of a directed polymer on a cylinder, to appear in SIAM Journ. Math. Anal., https://arxiv.org/abs/2207.14091

[3] A. Dunlap, Y. Gu, T. Komorowski„ Fluctuations of the KPZ equation on a large torus, to appear in Comm. on Pure and Applied Math., https://arxiv.org/abs/2111.03650

Abstract: In this talk, I will discuss recent results about the effective dynamics of interacting fermions in the mean-field scaling regime. In particular, I will show that under suitable assumptions on the interaction and for initial data close to quasi-free states with nonzero pairing obeying suitable semiclassical estimates, the validity of the Hartree-Fock-Bogoliubov (HFB) equation can be rigorously proved. More precisely, one proves that the k-body reduced density matrix can be computed from the solution of HFB via the Wick's rule, with an error that is rigorously bounded and becomes negligible as the average number of particles goes to infinity. The proof is built adapting the techniques used to derive the Hartree-Fock equation, that is the problem without pairing.

Joint work with Marcello Porta and Julien Sabin, arXiv:2310.15280.

Abstract: Scaling and criticality have always played a pivotal role in understanding fundamental properties of solutions to a given PDE. Its relevance is well-known in fluid dynamics, where scaling invariant (i.e. critical) spaces have nowadays become cornerstone tools for studying the Navier-Stokes equations. In recent years, a more systematic treatment and comprehensive theory for treating critical spaces have been developed. The aim of this talk is to discuss the basic features of the latter in the context of stochastic PDEs (SPDEs), where the stochasticity models turbulent phenomena. In particular, the $L^q$-setting of SPDEs with $q\gg 2$, pioneered by N.V. Krylov, plays a crucial role. We will explain the wide reach of this theory by means of its application to stochastic reaction diffusion equations, which arise in several physical and engineering applications such as chemical reactions and population dynamics. We will also highlight connections with regularization by noise results, and finally lay out some open problems.

Based on joint works with M. Veraar (TU Delft).

We derive a Pauli exclusion principle for extended fermion-based anyons of any positive radius and any non-trivial statistics parameter. We consider N 2D fermionic particles coupled to magnetic flux tubes of non-zero radius, and prove a Lieb-Thirring inequality for the corresponding many-body kinetic energy operator. The implied constant is independent of the radius of the flux tubes, and proportional to the statistics parameter.

Inspired by questions concerning the evolution of phase fields, we study the Allen-Cahn equation in dimension 2 with white noise initial datum. In a weak coupling regime, where the nonlinearity is damped in relation to the smoothing of the initial condition, we prove Gaussian fluctuations. The effective variance that appears can be described as the solution to an ODE. Our proof builds on a Wild expansion of the solution, which is controlled through precise combinatorial estimates. Joint works with Simon Gabriel, Martin Hairer, Khoa Lê and Nikos Zygouras. 

 Ever since Osterwalder and Schrader's pioneering work, it has beenestablished that Fermion systems can be described using Grassmann variables and Grassmann measures. In my talk, I will delve into this perspective, elucidating fundamental stochastic analytical tools, such as Grassmann Brownian motion and non-commutative Lebesgue spaces, which allow for a more thorough stochastic analytical characterisation of Grassmann measures. To exemplify the utility of this approach, I will discuss the advancements achieved in describing some prototypical interacting Grassmann measures.

Based on joint works with F. De Vecchi, M. Gordina and M. Gubinelli.

After reviewing the main aspect of high temperature superconductors, I will

stress how the recent high-resolution RIXS (resonant inelastic X-ray scattering)

experiments have given a new impulse to the physics of cuprates [1]. In

particular, the newly discovered short-range dynamical charge density

fluctuations with short correlation length [2], precursors of the three-

dimensional charge density waves, account for the long-standing problem of

the strange metal behaviour of the cuprates [3]. Due to their broadness,

charge density fluctuations mediate an almost isotropic scattering among the

fermi quasiparticles. Above the superconducting critical temperature and

around optimal doping, within a time-dependent Landau-Ginzburg approach,

isotropic scattering provides the famous strange metal behaviour and the

linear-in-T resistivity. This is realized for temperatures greater than the

characteristic energy (proportional to the inverse correlation length squared

and to the inverse dissipation parameter) [3]. However, the linearity in T of the

resistivity and a divergent specific heath [4], are observed down to the lowest

temperatures (the so called Planckian behavior) by experiments in the

presence of high pulsed magnetic field to suppress superconductivity. To

account of these results at low T, we must reduce the characteristic energy of

the fluctuations finite by increasing the damping while the correlation length

stays. Namely, we are suggesting a new paradigm [5] and [6] in contrast with

the standard criticality models. Usually, the diverging correlation length is

invoked to produce quantum criticality as the temperature goes to zero, here

instead the strange criticality with linear in T resistivity and a diverging specific

heat are accounted for by the increase of the damping parameter only. When

the damping increases by lowering the temperature, the charge density

fluctuations relax at longer and longer times giving rise to an anomalous

criticality for a glass of small islands of charge density fluctuations correlated in

time not in space.

References

[1] R. Arpaia & G. Ghiringhelli. Charge order at high temperature in cuprate superconductors.

J. Phys. Soc. Jpn. 90, 111005 (2021).

[2] R. Arpaia, S. Caprara, R. Fumagalli, G. De Vecchi, Y.Y. Peng, E. Andersson, D. Betto, G. M. De Luca,

N. B. Brookes, F. Lombardi, M. Salluzzo, L. Braicovich, C. Di Castro, M. Grilli, G. Ghiringhelli,

Dynamical charge density fluctuations pervading the phase diagram of a copper-based high-Tc

superconductor, Science 365, 906 (2019)

[3] G. Seibold, R. Arpaia, Y. Y. Peng, R. Fumagalli, L. Braicovich, C. Di Castro, M. Grilli, G. Ghiringhelli,

and S. Caprara, Strange metal behaviour from charge density fluctuations in cuprates, Commun.

Phys. 4, 7 (2021).

[4] Michon, et al, Thermodynamic signatures of quantum criticality in cuprate superconductors,

Nature 567, 218222 (2019)

[5] S. Caprara, C. Di Castro, G. Mirarchi, G. Seibold and M. Grilli

Dissipation-driven strange metal behavior, Commun. Phys. 5, 10 (2022).

[6] R. Arpaia et al. Signature of quantum criticality in cuprates by charge density fluctuations. Nat

Commun 14, 7198 (2023). https://doi.org/10.1038/s41467-023-42961-5

We consider a low-density Bose gas interacting through a repulsive potential in the thermodynamic limit. Using upper and lower bounds, we justify a Lee-Huang-Yang type formula for the free energy at suitably low temperatures. This is a joint work with F. Haberberger, P. T. Nam, B. Schlein, R. Seiringer and A. Triay. 

Systems of N particles, whose evolution is described by a mean-field pairwise interaction, and are driven by independent Brownian motions, are often expected to converge in the large N limit to solutions of a McKean-Vlasov equation, a non local SDE in the probability space. It is less known, but also classical, that the result is not specific to the drivers being Brownian; as long as the interaction kernel is smooth, the same holds true for a much larger class of processes, like fractional Brownian motion (fBm) of Hurst parameter $H\in (0,1)$. This raises the question of understanding what happens for more singular interactions. On one hand, fBm is not Markovian nor a martingale, so Ito calculus and PDE techniques are not available anymore; on the other, fBm is known to have a strong regularizing effect on standard SDEs, making their analysis possible for very rough, possibly distributional drifts. In this talk, I will present a propagation of chaos result, which holds for a general class of rough interaction kernels, whose regularity depends on the value $H$. The proofs rely on recently developed stochastic sewing techniques, which have been proved to be very effective in many contexts concerning S(P)DEs driven by (fractional) Brownian noise. Based on an ongoing joint work with K. Le and A. Mayorcas.

This talk is really three talks. In the first part, our main focus is an algorithm used to approximately solve an optimal transport problem that often arises in economic applications: how do we minimize the cost of matching producers with consumers, when the cost of a matching is a concave function of the distance? In the second part, we analyze how feedback can be used to improve the performance of certain card guessing games, closely related to well-known statistical problems such as bias-detection in clinical trials and hypothesis testing. In the third part, motivated by a novel notion of curvature on graphs, we provide some close-to-exact formula for the hitting times in Erdös-Rényi random graphs. These stories are held together by a common theme: sometimes, greediness and dependencies are not that harmful. This is based on joint works with J. He, S. Steinerberger and R. Tripathi. 

This talk is really three talks. In the first part, our main focus is an algorithm used to approximately solve an optimal transport problem that often arises in economic applications: how do we minimize the cost of matching producers with consumers, when the cost of a matching is a concave function of the distance? In the second part, we analyze how feedback can be used to improve the performance of certain card guessing games, closely related to well-known statistical problems such as bias-detection in clinical trials and hypothesis testing. In the third part, motivated by a novel notion of curvature on graphs, we provide some close-to-exact formula for the hitting times in Erdös-Rényi random graphs. These stories are held together by a common theme: sometimes, greediness and dependencies are not that harmful. This is based on joint works with J. He, S. Steinerberger and R. Tripathi. 

Two years ago, I talked about a quantitative lower bound that I obtained together with Steffen Polzer on the effective mass of the Polaron problem. At the time, it was the first quantitative lower bound after many years of no results. However, in the last two years, this result has been made largely obsolete. By two works of Sellke and Bazaes/Mukherjee/Sellke/Varadhan, the problem is now almost solved. I will recapitulate what we did at the time, and report about the new methods that are used, and how they relate to what was known before, and how they lead to the result. 

Two years ago, I talked about a quantitative lower bound that I obtained together with Steffen Polzer on the effective mass of the Polaron problem. At the time, it was the first quantitative lower bound after many years of no results. However, in the last two years, this result has been made largely obsolete. By two works of Sellke and Bazaes/Mukherjee/Sellke/Varadhan, the problem is now almost solved. I will recapitulate what we did at the time, and report about the new methods that are used, and how they relate to what was known before, and how they lead to the result. 

The goal of this talk  is to investigate the dynamics of semi-relativistic or non- relativistic charged particles in interaction with a scalar meson field. Our main contribution is the derivation of the classical dynamics of a particle-field system as an effective equation of the quantum microscopic Nelson model, in the classical limit where the value of the Planck constant approaches zero ($\hbar \rightarrow  0$). We use a Wigner measure approach to study the transition from quantum to classical dynamics. Then, as a consequence of this transition, we establish the global well-posedness of the classical particle-field interacting system.

The distribution of the temperature within a body where the transfer of heat is due only to radiative processes can be described by the radiative transfer equation (RTE). In this talk we will first introduce this kinetic equation, specifically we will consider the stationary RTE for absorption and scattering coefficients which can depend on both frequency and local temperature. We will reformulate the boundary value problem as an equivalent fixed-point equation for the temperature T for which an existence theory will be developed. In particular we will study the compactness for the resulting non-linear problem involving exponentials of some integrals along straight lines.

If time permits besides the pure absorption/emission case we will study also the full equation with both absorption/emission and scattering terms for which the compactness result has to be combined with the definition of suitable Green functions. 

This is joint work with Jin Woo Jang and Juan J.L. Velázquez.