The partition function of Beta-ensembles with complex potentials

• Alex Little (ENS Lyon)

A Beta-ensemble can be viewed as a gas of particles confined to a line with logarithmic pairwise interactions and at inverse temperature Beta. The partition function of a Beta-ensemble involves a large parameter N which appears both in the integrand and as the number of integrations, and thus its asymptotic analysis could be regarded as an infinite-dimensional version of the Laplace method. In joint work with A. Guionnet and K. Kozlowski, we consider the partition function of a Beta-ensemble with a complex-valued potential. We prove, under certain hypotheses, a full 1/N expansion of this partition function and explicitly identify the first few terms. Because the integrand is complex, and hence oscillatory, our method could be regarded as an infinite-dimensional version of the Steepest Descent method. ArXiv reference: https://arxiv.org/abs/2411.10610

Asymptotics of the two-dimensional Yang-Mills partition function

• Thibaut Lemoine (Collège de France)

The two-dimensional Yang-Mills measure is a probabilistic model describing the quantum Euclidean Yang-Mills theory in two dimensions. I will briefly describe the construction of this measure, first in lattice gauge theory, then in the continuum, revisiting a construction by Lévy (2003, 2010), then I will focus on its partition function on compact surfaces, with structure group the unitary group U(N). We will see what happens for the partition function in the large N regime, akin to random matrix theory, and in particular we will describe a so-called gauge/string duality on a torus, confirming predictions in theoretical physics by Gross and Taylor (1993). Based on a joint work with Mylène Maïda (Université de Lille).