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Yves Le Jan

Publications and source records attributed to Yves Le Jan.

At least 19 recordsLinked to original sources
arXiv2026

Sampling biased spanning forests through marked vertices

We study biased spanning forests on the complete graph \(K_N\), obtained by assigning weight proportional to \(κ^q\) to spanning trees on \(K_N\cup\{Δ\}\), where \(q\) is the degree of a distinguished root \(Δ\). Fixing a finite set \(L\) of marked vertices, we analyze the minimal \(Δ\)-rooted subtree connecting \(L\) to \(Δ\). In this sense, we investigate the effect of partially sampling a large random spanning forest through finitely many vertices. For fixed \(κ\), the reduced subtree is asymptotically a uniformly distributed binary tree, and graph distances rescaled by \(\sqrt N\) converge jointly in distribution to the explicit limit introduced by Aldous in his study of the Brownian CRT. We show that the scale \(κ\asymp\sqrt N\) is critical: if \(κ=o(\sqrt N)\), the marked vertices lie asymptotically in a single component; if \(κ\gg\sqrt N\), the induced partition is asymptotically discrete and the distances to \(Δ\) are negligible on the \(\sqrt N\) scale. In the critical regime \(κ=c\sqrt N\), the induced partition of the marked vertices converges to a non-degenerate limit law, namely the \(\tfrac12\)-stable Poisson--Kingman partition studied by Pitman. We also describe a continuous-time edge-cutting dynamics: under the critical scaling, its fixed-time marginals are obtained by the parameter shift \(c\mapsto c+a\). Our approach remains entirely discrete and combinatorial: the asymptotics are obtained from explicit determinant formulas and elementary expansions, without invoking continuum limits.

math.CO↗
arXiv2026

Genetic contribution of advantaged ancestors in the biparental Moran model -- finite selection

We study a population of $N$ individuals evolving according to a biparental Moran model with two types, one being advantaged compared to the other. The advantage is conferred by a Mendelian mutation, which reduces the death probability of individuals carrying it. We assume that a proportion $a$ of individuals initially carry this mutation, which therefore eventually gets fixed with high probability. After a long time, we sample a gene uniformly from the population, at a new locus, independent of the locus under selection, and calculate the probability that this gene originated from one of the initially advantaged individuals, when the population size is large. Our theorem provides quantitative insights, such as the observation that under strong viability selection, if only $1\%$ of the individuals are initially advantaged, up to $19\%$ of the population's genome will originate from them after a long time.

math.PR↗
arXiv2024

Genetic contribution of an advantaged mutant in the biparental Moran model -- finite selection

We consider a population of N individuals, whose dynamics through time is represented by a biparental Moran model with two types: an advantaged type and a disadvantaged type. The advantage is due to a mutation, transmitted in a Mendelian way from parent to child that reduces the death probability of individuals carrying it. We assume that initially this mutation is carried by a proportion a of individuals in the population. Once the mutation is fixed, a gene is sampled uniformly in the population, at a locus independent of the locus under selection. We then give the probability that this gene initially comes from an advantaged individual, i.e. the genetic contribution of these individuals, as a function of a and when the population size is large.

math.PR↗
arXiv2022

Connections and loops intertwinning

On a finite graph, we prove that trace of holonomies determine an intertwining relation between merge-and-split generators on collections of geodesic loops ensembles and Casimir operators on unitary connections. By adding a deformation part to the generator on loops, this result is extended to the Casimir operator modified in order to be self adjoint with respect to Yang- Mills measure.

math.PR↗
arXiv2022

Genetic contribution of an advantaged mutant in the biparental Moran model

We consider a population of haploid individuals reproducing sexually, i.e. for which the genome of each individual is a random mixture of the genome of its two parents. We assume that initially one individual carries a mutation at one locus, and that individuals carrying this mutation have an advantage regarding genome transmission. Our aim is to study the long time effect of this mutation on the genetic composition of the population, when population size is large.

math.PR↗
arXiv2022

Genetics of the biparental Moran model

Our goal is to study the genetic composition of a population in which each individual has 2 parents, who contribute equally to the genome of their ospring. We use a biparental Moran model, which is characterized by its xed number N of individuals. We x an individual and consider the proportions of the genomes of all individuals living n time steps later, that come from this individual. We rst prove that when n goes to innity, these proportions all converge almost surely towards the same random variable. We then rigorously prove that when N then goes to innity, this random variable multiplied by N (i.e. the stationary weight of any ancestor in the whole population) converges in law towards the mixture of a Dirac measure in 0 and an exponential law with parameter 1/2, and that the weights of several given ancestors are independent.

math.PR↗
arXiv2020

On discrete loop signatures and Markov loops topology

Our purpose is to explore, in the context of loop ensembles on finite graphs, the relations between combinatorial group theory, loops topology, loop measures, and signatures of discrete paths. We determine the distributions of the loop homotopy class, and of the first and second homologies, defined by the lower central series of the fundamental group.

math.PR↗
arXiv2020

Limit theorems for loop soup random variables

This article deals with limit theorems for certain loop variables for loop soups whose intensity approaches infinity. We first consider random walk loop soups on finite graphs and obtain a central limit theorem when the loop variable is the sum over all loops of the integral of each loop against a given one-form on the graph. An extension of this result to the noncommutative case of loop holonomies is also discussed. As an application of the first result, we derive a central limit theorem for windings of loops around the faces of a planar graphs. More precisely, we show that the winding field generated by a random walk loop soup, when appropriately normalized, has a Gaussian limit as the loop soup intensity tends to $\infty$, and we give an explicit formula for the covariance kernel of the limiting field. We also derive a Spitzer-type law for windings of the Brownian loop soup, i.e., we show that the total winding around a point of all loops of diameter larger than $δ$, when multiplied by $1/\logδ$, converges in distribution to a Cauchy random variable as $δ\to 0$.

math.PR↗
arXiv2019

Equivariant diffusions on Principal bundles

Given a pair of second order diffusion operators, one on the total space of a principle bundle $N$ and the other on the base space $M$, intertwined by the projection $π:N\to M$, if the operator ${\mathcal A}$ on the base manifold has constant rank, we define a semi-connection on the principal bundle which allows to split the diffusion operator ${\mathcal B}$ on the total space into the sum of the horizontal lift of ${\mathcal A}$ and the other vertical. This allow to conclude a disintegration theorem for the law of ${\mathcal B}$. As an application, a decomposition of stochastic flow is given.

math.PR↗
arXiv2019

Integration by parts formulae for degenerate diffusion measures on path spaces and diffeomorphism groups

Integration by parts formulae are given for a class of measures on the space of paths of a smooth manifold $M$ determined by the laws of degenerate diffusions. The mother of such formulae, on the path space of diffeomorphism group of $M$ is shown to arise from a quasi-invariance property of measures determined by stochastic flows. From this the other formulae are derived by filtering out redundant noise using an associated LeJan-Watanabe connection.

math.PR↗
arXiv2019

Flows, coalescence and noise

We are interested in stationary "fluid" random evolutions with independent increments. Under some mild assumptions, we show they are solutions of a stochastic differential equation (SDE). There are situations where these evolutions are not described by flows of diffeomorphisms, but by coalescing flows or by flows of probability kernels. In an intermediate phase, for which there exist a coalescing flow and a flow of kernels solution of the SDE, a classification is given: All solutions of the SDE can be obtained by filtering a coalescing motion with respect to a subnoise containing the Gaussian part of its noise. Thus, the coalescing motion cannot be described by a white noise.

math.PR↗
arXiv2019

Brownian winding fields

The purpose of the present note is to review and improve the convergence of the renormalized winding fields.

math.PR↗
arXiv2019

Loop interactions and their representations in Fock space

Given a weighted graph, we show how to define two types of natural interactions which correspond to local interactions between two Fock spaces. The first type of interaction involves loop ensembles and spanning trees. The second type of interaction involves loop holonomies and random connections.

math.PR↗
arXiv2017

Markov loops topology

Since the work of Lawler and Werner on "loop soups", these ensembles have also been the object of many investigations. Their properties can be studied in the context of rather general Markov processes, in particular Markov chains on graphs. The purpose of the present work is to explore their topological properties.

math.PR↗
arXiv2017

Homology of Brownian loops

The purpose of this note is to extend to Brownian loops some homology and holonomy results obtained in the case of discrete loops on a graph

math.PR↗
arXiv2016

Markov loops, coverings and fields

We investigate the relations between the Poissonnian loop ensembles , their occupation fields, non ramified Galois coverings of a graph, the associated gauge fields, and random Eulerian networks.

math.PR↗