Week 17.03.2024 – 23.03.2024

The six-vertex model is an important toy-model in statistical mechanics for two-dimensional ice with a natural parameter D. When D=0, the so-called free-fermion point, the model is in natural correspondence with domino tilings of the Aztec diamond. Although this model is integrable for all D, there has been very little progress in understanding its statistics in the scaling limit for other values. In this talk, we focus on the six-vertex model with domain wall boundary conditions at D = 1/2, where it corresponds to alternating sign matrices (ASMs). We consider the level lines in a height function representation of ASMs. We report that the maximum of the topmost level line for a uniformly random ASMs has the GOE Tracy-Widom distribution after appropriate rescaling and will discuss many open problems related to this model. Much of this talk is based on joint work with Arvind Ayyer and Kurt Johansson.

We propose the Gauge Theory Bootstrap, a method to compute the pion S-matrix that describes the strongly coupled low energy physics of QCD and other similar gauge theories. The method looks for the most general S-matrix that matches at low energy the tree level amplitudes of the non-linear sigma model and at high energy, QCD sum rules and form factors. We compute pion scattering phase shifts for all partial waves with angular momentum $\ell<=3$ up to 2 GeV and calculate the low energy ChiPT coefficients. This is a theoretical/numerical calculation that uses as only data the pion mass $m_\pi$, pion decay constant $f_{\pi}$ and the QCD parameters $N_c=3$, $N_f=2$, $m_q$ and $\alpha_s$. All results are in reasonable agreement with experiment. In particular, we find the $\rho(770)$, $f_2(1270)$ and $\rho(1450)$ resonances and some initial indication of particle production near the resonances. The interplay between the UV gauge theory and low energy pion physics is an example of a general situation where we know the microscopic theory as well as the effective theory of long wavelength fluctuations but we want to solve the strongly coupled dynamics at intermediate energies. The bootstrap builds a bridge between the low and high energy by determining the consistent S-matrix that matches both and provides, in this case, a new direction to understand the strongly coupled physics of gauge theories. Based on work with Martin Kruczenski.

The Widom conjecture concerns the asymptotic spectral density of Toeplitz operators of the form $\Pi_U F \Pi_V F^* \Pi_U$, where $\Pi_U$ is the operator of multiplication by the indicator of an open set $U$ and $F$ is the Fourier transform, in a semclassical limit where the size of $U$ and/or $V$ tends to infinity. This conjecture was proved by Widom himself in the 80's and by A. Sobolev and his collaborators a decade ago.

Widom's initial motivation was to prove an analogue of a theorem by Basor on large Toeplitz matrices with indicator symbols, and in both cases one can translate the spectral asymptotics into probabilistic quantities for natural point process models -- for instance, Basor's result describes the number of eigenvalues of a random large unitary matrix which lie inside an interval of the unit circle.

In turns, this interpretation prompts potential generalisations of the Widom conjecture to operators built with other kinds of projectors, such as general spectral projectors for quantum hamiltonians. In this talk, I will present an overview of the Widom conjecture, the probabilistic interpretation, and my joint work with Gaultier Lambert (some of it in progress) towards the generalised Widom conjecture.

Kernel Stein discrepancies (KSDs) measure the quality of a
distributional approximation and can be computed even when the target
density has an intractable normalizing constant. Notable applications
include the diagnosis of approximate MCMC samplers and goodness-of-fit
tests for unnormalized statistical models. The present work analyzes
the convergence control properties of KSDs. We first show that
standard KSDs used for weak convergence control fail to control moment
convergence. To address this limitation, we next provide sufficient
conditions under which alternative diffusion KSDs control both moment
and weak convergence. As an immediate consequence we develop, for each
q>0, the first KSDs known to exactly characterize q-Wasserstein
convergence.