# 21-22

November , 22 2021 at 17,30 - Cladio Landim

(Ruen- France, IMPA Brazil)

Title: Steady state large deviations for one-dimensional, symmetric exclusion processes in weak contact with reservoirs.

### Lorenzo Bertini

### (Roma I -La Sapienza- University of Rome )

On the probability of observing energy increasing solutions to the Boltzmann equation.

Date : December 20, at 14:00 (italian time)

The seminar will be held in presence in the GSSI-Main Lecture Hall, and streamed through zoom

To attend the seminar in person, we kindly invite you to register at the following page: https://smaq.mat-univaq.com/booking.

ABSTRACT

Weak solutions to the homogeneous Boltzmann equation with increasing energy have been constructed by Lu and Wennberg. We consider an underlying microscopic stochastic model with binary collisions and show that these solutions are atypical.

More precisely, we prove that the probability of observing these paths is exponentially small in the number of particles and compute the exponential rate.

Spontaneous magnetization in the classical 3D Heisenberg model: asymptotic nature of the low temperature expansion .

SMAQ Seminar & Conferenza del Centro Linceo.

Date : February 8, 2022 at 15:00 (italian time)

The seminar will be held in presence in the GSSI-Main Lecture Hall, and streamed through zoom

To attend the seminar in person, we kindly invite you to register at the following page: https://smaq.mat-univaq.com/booking.

ABSTRACT

In 1976, in one of the most influential papers in mathematical statistical mechanics, Frohlich-Simon-Spencer (FSS) proved the existence of orientational long range order for the classical XY and Heisenberg models in three or more dimensions. In 1981, Bricmont-Fontaine-Lebowitz-Lieb-Spencer (BFLLS) extended the FSS result for the 3D rotator model, proving that the formal low temperature expansion for its magnetization is asymptotic; i.e., any truncation at finite order n of such a formal series is close to the actual magnetization up to an error o(T^n) at low temperatures. The generalisation of this result to the Heisenberg model remained open since, for a difficulty to extend the BFLLS analysis to the case of non-abelian rotational symmetry. Recently, in collaboration with S. Ott, we succeeded in overcoming some of the technical limitations of the BFLLS proof, preventing them to apply their ideas to the Heisenberg model. In this talk I will report the statement of our main result on the asymptotic nature of the low-temperature expansion for the spontaneous magnetization of the 3D Heisenberg model and discuss the main ideas of the proof.

### (TU Darmstadt)

A lower bound for the effective mass of the Fröhlich polaron at strong coupling.

SMAQ Seminar & Conferenza del Centro Linceo.

Date : March 14, 2022 at 14:300 (italian time)

The seminar will be held in presence in the GSSI-Main Lecture Hall, and streamed through zoom

To attend the seminar in person, we kindly invite you to register at the following page: https://smaq.mat-univaq.com/booking.

## Abstract

The Fröhlich polaron describes a quantum particle in a polar crystal. By the interaction with the field modes of the crystal, the particle appears heavier than it is, that is referred to as effective mass. An old conjecture of Landau and Pekar is that this effective mass should scale like $\alpha^4$ in the coupling parameter $\alpha$ that measures the strength of the particle/field interaction. It has been proved recently by Lieb and Seiringer that the effective mass diverges as $\alpha \to \infty$, but there were no quantitative lower bounds.

I will present new results obtained jointly with Steffen Polzer (Geneva) that go 10% of the way towards the Landau/Pekar conjecture: we show that the effective mass is bounded below by a constant time $\alpha^{2/5}$. The method uses a recent representation of the Polaron path measure due to Mukherjee and Varadhan, describing the polaron as a statistical mechanics system of interacting intervals on the real line.

### (Sapienza, Università di Roma)

Percolative metal-to-superconductor transition in superconductors with disorder at the nanoscale

Date : April 4, 2022 at 14:30 (italian time)

The seminar will be held in presence in the GSSI-Main Lecture Hall, and streamed through zoom

onsite: GSSI-Main Lecture Hall booking form

zoom: https://us02web.zoom.us/j/87571297606

## Abstract

After introducing some experimental evidence for the occurrence of a percolative metal-to-superconductor transition in superconductors with disorder at the nanoscale, I will discuss the theory of transport in random media, and its description within the effective medium theory. Finally, I will compare the results obtained within the effective medium theory with those obtained by means of the numerical solution of random resistor networks, in the absence and in the presence of spatial correlations within the superconducting cluster.

Information. This seminar is part of the Stochastic Modelling in l'Aquila (SMAQ) seminar series, jointly organised with the University of L'Aquila, and one of the Conferences of the Centro Linceo (link).

### (Georgia Institute of Technology)

Some results on a simple model of kinetic theory

Date : June 20, 2022 at 14:30 (italian time)

The seminar will be held in presence in the GSSI-Main Lecture Hall, and streamed through zoom

onsite: GSSI-Auditorium booking form

zoom: https://us02web.zoom.us/j/87571297606

## Abstract

A two-layer quasi-geostrophic (2LQG) model for geophysical flows is studied, with the upper layer being perturbed by additive noise. This model is popular in the geosciences, for instance to study the effects of a stochastic wind forcing on the ocean. A rigorous mathematical analysis however meets with the challenge that the noise configuration is spatially degenerate as the stochastic forcing acts only on the top layer. We will discuss the problem of unique ergodicity and exponential convergence of transition probabilities as well as response theory for stochastic partial differential equations like the 2LQG model.