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We define two natural families of determinantal random subgraphs of a finite connected graph, one supported by acyclic spanning subgraphs (spanning forests) with fixed number of components, the other by connected spanning subgraphs with fixed number of independent cycles. Each family generalizes the uniform spanning tree and the generating functions of these probability measures generalize the classical Kirchhoff and Symanzik polynomials. We emphasize the matroidal nature of this construction, as well as possible generalisations., Comment: 41 pages, 5 figures

In this short note, we extend to the continuous case a mean projection theorem for discrete determinantal point processes associated with a finite range projection, thus strengthening a known result in random linear algebra due to Ermakov and Zolotukhin. We also give a new formula for the variance of the exterior power of the random projection., Comment: 7 pages

These notes, echoing a conference given at the Strasbourg-Zurich seminar in October 2017, are written to serve as an introduction to 2-dimensional quantum Yang-Mills theory and to the results obtained in the last five to ten years about its so-called large N limit., Comment: 40 pages, 12 figures

We introduce and study a class of determinantal probability measures generalising the class of discrete determinantal point processes. These measures live on the Grassmannian of a real, complex, or quaternionic inner product space that is split into pairwise orthogonal finite-dimensional subspaces. They are determined by a positive self-adjoint contraction of the inner product space, in a way that is equivariant under the action of the group of isometries that preserve the splitting., Comment: 55 pages, 2 figures

In this short note, we revisit Zeilberger's proof of the classical matrix-tree theorem and give a unified concise proof of variants of this theorem, some known and some new., Comment: 10 pages, 2 figures, and 1 colour

Classical isomorphism theorems due to Dynkin, Eisenbaum, Le Jan, and Sznitman establish equalities between the correlation functions or distributions of occupation times of random paths or ensembles of paths and Markovian fields, such as the discrete Gaussian free field. We extend these results to the case of real, complex, or quaternionic vector bundles of arbitrary rank over graphs endowed with a connection, by providing distributional identities between functionals of the Gaussian free vector field and holonomies of random paths. As an application, we give a formula for computing moments of a large class of random, in general non-Gaussian, fields in terms of holonomies of random paths with respect to an annealed random gauge field, in the spirit of Symanzik's foundational work on the subject., Comment: 51 pages, 10 figures. This version contains a new introduction, an additional Section (6.8) detailing an important example (the case of trace-positive holonomies), and a treatment of the quaternionic case. The introductory material on continuous time random walks on multigraphs in Section 1 was also simplified

We give a rigorous proof of the fact that a phase transition discovered by Douglas and Kazakov in 1993 in the context of two-dimensional gauge theories occurs. This phase transition can be formulated in terms of the Brownian bridge on the unitary group U(N) when N tends to infinity. We explain how it can be understood by considering the asymptotic behaviour of the eigenvalues of the unitary Brownian bridge, and how it can be technically approached by means of Fourier analysis on the unitary group. Moreover, we advertise some more or less classical methods for solving certain minimisation problems which play a fundamental role in the study of the phase transition.

We study the large N asymptotics of the Brownian motions on the orthogonal, unitary and symplectic groups, extend the convergence in non-commutative distribution originally obtained by Biane for the unitary Brownian motion to the orthogonal and symplectic cases, and derive explicit estimates for the speed of convergence in non-commutative distribution of arbitrary words in independent Brownian motions. Using these results, we construct and study the large N limit of the Yang-Mills measure on the Euclidean plane with orthogonal, unitary and symplectic structure groups. We prove that each Wilson loop converges in probability towards a deterministic limit, and that its expectation converges to the same limit at a speed which is controlled explicitly by the length of the loop. In the course of this study, we reprove and mildly generalise a result of Hambly and Lyons on the set of tree-like rectifiable paths. Finally, we establish rigorously, both for finite N and in the large N limit, the Schwinger-Dyson equations for the expectations of Wilson loops, which in this context are called the Makeenko-Migdal equations. We study how these equations allow one to compute recursively the expectation of a Wilson loop as a component of the solution of a differential system with respect to the areas of the faces delimited by the loop.

We give a bijective proof of the fact that the number of k-prefixes of minimal factorisations of the n-cycle (1...n) as a product of n-1 transpositions is n^{k-1}\binom{n}{k+1}. Rather than a bijection, we construct a surjection with fibres of constant size. This surjection is inspired by a bijection exhibited by Stanley between minimal factorisations of an n-cycle and parking functions, and by a counting argument for parking functions due to Pollak.

We prove that for a finite collection of real-valued functions $f_{1},...,f_{n}$ on the group of complex numbers of modulus 1 which are derivable with Lipschitz continuous derivative, the distribution of $(\tr f_{1},...,\tr f_{n})$ under the properly scaled heat kernel measure at a given time on the unitary group $\U(N)$ has Gaussian fluctuations as $N$ tends to infinity, with a covariance for which we give a formula and which is of order $N^{-1}$. In the limit where the time tends to infinity, we prove that this covariance converges to that obtained by P. Diaconis and S. Evans in a previous work on uniformly distributed unitary matrices. Finally, we discuss some combinatorial aspects of our results., Comment: 44 pages

In this paper, we investigate a continuous family of notions of independence which interpolates between the classical and free ones for non-commutative random variables. These notions are related to the liberation process introduced by D. Voiculescu. To each notion of independence correspond new convolutions of probability measures, for which we establish formulae and of which we compute simple examples. We prove that there exists no reasonable analogue of classical and free cumulants associated to these notions of independence., Comment: 27 pages

We define a notion of Markov process indexed by curves drawn on a compact surface and taking its values in a compact Lie group. We call such a process a two-dimensional Markovian holonomy field. The prototype of this class of processes, and the only one to have been constructed before the present work, is the canonical process under the Yang-Mills measure, first defined by Ambar Sengupta and later by the author . The Yang-Mills measure sits in the class of Markovian holonomy fields very much like the Brownian motion in the class of Levy processes. We prove that every regular Markovian holonomy field determines a Levy process of a certain class on the Lie group in which it takes its values, and construct, for each Levy process in this class, a Markovian holonomy field to which it is associated. When the Lie group is in fact a finite group, we give an alternative construction of this Markovian holonomy field as the monodromy of a random ramified principal bundle.

We establish a convergent power series expansion for the expectation of a product of traces of powers of a random unitary matrix under the heat kernel measure. These expectations turn out to be the generating series of certain paths in the Cayley graph of the symmetric group. We then compute the asymptotic distribution of a random unitary matrix under the heat kernel measure on the unitary group U(N) as N tends to infinity, and prove a result of asymptotic freeness result for independent large unitary matrices, thus recovering results obtained previously by Xu and Biane. We give an interpretation of our main expansion in terms of random ramified coverings of a disk. Our approach is based on the Schur-Weyl duality and we extend some of our results to the orthogonal and symplectic cases.

We construct one Yang-Mills measure on a compact surface for each isomorphism class of principal bundles over this surface. For this, we define a new discrete gauge theory which is essentially a covering of the usual one. We prove that the measures correponding to different isomorphism classes of bundles or to different total areas of the surface are mutually singular. We give also a combinatorial computation of the partition functions based on the formalism of fat graphs.

We prove the first mathematical result relating the Yang-Mills measure on a compact surface and the Yang-Mills energy. We show that, at the small volume limit, the Yang-Mills measures satisfy a large deviation principle with a rate function which is expressed in a simple and natural way in terms of the Yang-Mills energy.

When G is a product of orthogonal, unitary and symplectic groups, we show that the Wilson loops generate a dense subalgebra of continuous observables on the configuration space of lattice gauge theory with structure group G.

We construct and study the Yang-Mills measure in two dimensions. According to the informal description given by the physicists, it is a probability measure on the space of connections modulo gauge transformations on a principal bundle with compact structure group. We are interested in the case where the base space of this bundle is a compact orientable surface. The construction of the measure in a discrete setting, where the base space of the fiber bundle is replaced by a graph traced on a surface, is quite well understood thanks to the work of E. Witten. In contrast, the continuum limit of this construction, which should allow to put a genuine manifold as base space, still remains problematic. This work presents a complete and unified approach of the discrete theory and of its continuum limit. We give a geometrically consistent definition of the Yang-Mills measure, under the form of a random holonomy along a wide, intrinsic and natural class of loops. This definition allows us to study combinatorial properties of the measure, like its Markovian behaviour under the surgery of surfaces, as well as properties specific to the continuous setting, for example, some of its microscopic properties. In particular, we clarify the links between the Yang-Mills measure and the white noise and show that there is a major difference between the Abelian and semi-simple theories. We prove that it is possible to construct a white noise using the measure as a starting point and vice versa in the Abelian case but we show a result of asymptotic independence in the semi-simple case which suggests that it is impossible to extract a white noise from the measure., Comment: LaTeX, 113 pp, 12 figures