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Thierry Lévy

Publications and source records attributed to Thierry Lévy.

17 recordsLinked to original sources
arXiv2026

Determinantal random subgraphs

We define two families of determinantal random spanning subgraphs of a finite connected graph, one supported by acyclic spanning subgraphs (spanning forests) with fixed number of connected 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 call Symanzik spanning forests the elements of the acyclic spanning subgraphs family, and single out a particular determinantal mixture of these, having as kernel a normalized Laplacian on $1$-forms, which we call the Laplacian spanning forest. Our proofs rely on a set of integral and real or complex (which we call geometric) multilinear identies involving cycles, coboundaries, and forests on graphs. We prove these identities using classical pieces of the algebraic topology of graphs and the exterior calculus applied to finite determinantal point processes, both of which we treat in a self-contained way. We emphasize the matroidal nature of our constructions, thereby showing how the above two families of random spanning subgraphs are dual to one another, as well as possible generalisations.

math.PR↗
arXiv2026

Trace identities for quiver representations

We give an expression for the determinant of the twisted Laplacian associated with any linear representation of a finite quiver in terms of traces of the holonomy of its cycles. To establish this expression, we prove a general identity for the determinant of a block matrix in terms of traces of products of its blocks. We give two proofs, one purely enumerative and one using generating series. In the special case of a finite graph equipped with a vector bundle and a connection, the twisted Laplacian determinant admits a combinatorial interpretation as a weighted count of tuples of oriented cycle-rooted spanning forests, where the weights involve traces of holonomies along cycles formed by combining the edges of the forests.

math.CO↗
arXiv2026

A positive formula for volumes of moduli spaces of flat unitary connections on compact surfaces

We provide a manifestly positive expression for the volume of the moduli spaces of flat $\mathrm{U}(n)$-valued connections on punctured compact oriented surfaces. This volume is obtained by summing volumes of explicit polytopes describing coloured honeycombs on a polygon, in the spirit of the work of Knutson and Tao describing the spectrum of the sum of two hermitian matrices. As a corollary, we also provide a positive formula for marginals of the $\mathrm{U}(n)$-valued Yang-Mills measure on a compact oriented surface in terms of the probability distribution of an explicit path process.

math.PR↗
arXiv2026

A combinatorial formula for Wilson loop expectations on compact surfaces

We give an almost purely combinatorial expression for Wilson loop expectations of the Yang-Mills holonomy process with values in the unitary group on a compact oriented surface, possibly with boundary and arbitrary boundary conditions. Our main result computes the non-normalized expectation of products of traces of holonomies along an arbitrary family of immersed curves with transverse self-intersections and no triple points. It is expressed as a sum over assignments of highest weights of the unitary group to the connected components of the complement of the curves. Each term is a product of a Gaussian exponential factor, dimensions of unitary representations, and local contributions at the intersection points given by the sine or cosine of an angle determined by the surrounding highest weights. As an application, we obtain a new and short proof of the Makeenko-Migdal equations on arbitrary compact surfaces.

math.PR↗
arXiv2023

On the mean projection theorem for determinantal point processes

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.

math.PR↗
arXiv2020

Determinantal probability measures on Grassmannians

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.

math.PR↗
arXiv2020

Covariant Symanzik identities

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.

math.PR↗
arXiv2019

A colourful path to matrix-tree theorems

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.

math.CO↗
arXiv2015

On The Douglas-Kazakov Phase Transition

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.

math.PR↗
arXiv2012

The master field on the plane

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.

math-ph↗
arXiv2011

Central limit theorem for the heat kernel measure on the unitary group

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.

math.PR↗
arXiv2011

Prefixes of minimal factorisations of a cycle

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.

math.CO↗
arXiv2008

A continuous semigroup of notions of independence between the classical and the free one

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.

math.PR↗
arXiv2008

Two-dimensional Markovian holonomy fields

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.

math.PR↗
arXiv2008

Schur-Weyl duality and the heat kernel measure on the unitary group

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.

math.PR↗
arXiv2005

Large deviations for the Yang-Mills measure on a compact surface

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.

math-ph↗