This is a joint work with Laurent MÃ©nard and Gilles Schaeffer.

I will show how to compute the eigenvalues and eigenvectors of a sum of two random matrices. This involves free probability and analytic functions, but I will not assume any knowledge of free probability. Then I will study repeated sums of random matrices and investigate the time homogeneous case, where some nice conformal mappings of the upper half plane appear.

We study the local limits of uniform triangulations chosen uniformly over those with fixed size and genus in the regime where the genus is proportional to the size. We show that they converge to the Planar Stochastic Hyperbolic Triangulations introduced by Curien. This generalizes the convergence of uniform planar triangulations to the UIPT of Angel and Schramm, and proves a conjecture of Benjamini and Curien. As a consequence, we obtain new asymptotics on the enumeration of high genus triangulations. This is a joint work with Baptiste Louf.

We introduce fully simple maps as maps with non-intersecting disjoint boundaries. It is well-known that the generating series of ordinary maps satisfy a universal recursive procedure, called topological recursion (TR). We propose a combinatorial interpretation of the important and still mysterious symplectic transformation which exchanges `x` and `y` in the initial data of the TR (the spectral curve). We give elegant formulas for the disk and cylinder topologies which recover relations already known in the context of free probability. For genus zero we provide an enumerative geometric interpretation of the so-called higher order free cumulants, which suggests the possibility of a general theory of approximate higher order free cumulants taking into account the higher genus amplitudes. We finally explain a universal relation between possibly disconnected fully simple and ordinary maps through double monotone Hurwitz numbers, which can be proved either using matrix models or bijective combinatorics.

Classical isomorphism theorems relate the local times of simple random walk to the square of a Gaussian free field. These theorems have been used to investigate many probabilistic questions, including cover times, self-avoiding walks, field theories, and large deviations for local times.

I will discuss new isomorphism theorems that relate reinforced random walks (the vertex-reinforced and vertex-diminished jump processes) to non-Gaussian spin systems that take values in hyperbolic and hemispherical spaces. There are also supersymmetric versions of these theorems. To illustrate the use of these theorems I will outline how they can be used to prove the vertex-reinforced jump process is recurrent in two dimensions.

Based on joint work with Roland Bauerschmidt and Andrew Swan.

The classical Peter–Weyl theorem asserts that the characters of the irreducible representations of a compact Lie group form an orthonormal basis of the space of square integrable and conjugation invariant functions on this group endowed with its normalised Haar measure. In the case of the group of unimodular complex numbers, this result is the foundation of the theory of Fourier series.

Given a compact Lie group, one can associate to every finite graph a space of gauge configurations, which is a certain Cartesian power of the group, on which acts another Cartesian power of the same group, called the gauge group. The action of the gauge group on the space of gauge configurations generalises the action of the group on itself by conjugation. In this context, the representations of the Lie group can also be used to produce an orthonormal basis of the space of square integrable and gauge-invariant functions on the space of gauge configurations : characters must be replaced by spin networks, a class of functions introduced by Penrose around 1970.

In this talk, I will explain what spin networks are, how they can be used to produce orthonormal families of gauge-invariant observables, and how they relate to another important family of gauge-invariant observables called Wilson loops. In doing so, I will present old results, and an open problem. I will then report on much more recent (in fact still ongoing) joint work with Adrien Kassel on the determinant of the covariant Laplacian.