Abstracts

Natasha Blitvic,

• Title : Permutations, moments, measures

• Abstract :  Which combinatorial sequences are moments of probability measures on the real line? Multiparameter continued fractions provide a surprisingly convenient answer, provided one can interpret them combinatorially. We will give an interpretation of one such continued fraction and deduce from it a wealth of corollaries. We will use this lens to explore the apparent link between positivity and hard open problems in permutation patterns. Based on a joint work with Einar Steingrimsson and, if time permits, more recent work with Einar Steingrimsson and Slim Kammoun.

Charles Bordenave,

• Title : Random permutations, random group actions and strong convergence

• Abstract :  In this talk, we will present some results on the convergence in operator norm of polynomials in random permutation matrices.

  Based on a joint work with Benoit Collins and another with Yanqi Qiu and Yiwei Zhang.

Nicolas Curien,

• Title : Universality for random surfaces in unconstrained genus

• Abstract :  Starting from an arbitrary sequence of polygons whose total perimeter is 2n, we can build an (oriented) surface by pairing their sides in a uniform fashion. Chmutov and Pittel (arXiv:1503.01816) have shown that, regardless of the configuration of polygons we started with, the degree sequence of the graph obtained this way is remarkably constant in total variation distance and converges towards a Poisson --Dirichlet partition as n→∞. We actually show that several other geometric properties of the graph are universal. This will highlight the link between random maps and random permutations.

  Based on joint work with Thomas Budzinski and Bram Petri.

Persi Diaconis (remote talk),

• Title : Enumeration by double cosets

• Abstract : 'Random matrix theory' has come to be the study of the eigenvalues of various ensembles of random matrices. The eigenvalues determine the conjugacy classes and of course the study of permutations by cycle type is a healthy part of combinatorial probability. There is a generalization that leads to some nice questions and results.: Let G be a compact group with (closed) subgroups H and K the equivalence relation s equivalent to t if s=htk (s and t in G, h in H, k in K) splits G into double cosets. One may ask 'pick g in G uniformly, what double coset is it likely to be in? Special cases of this set up give familiar objects: If G=S_n , H = S(lambda), K= S(mu) (parabolic subgroups) the double cosets are indexed by statisticians 'contingency tables' (arrays of non-negative integers with row sums lambda and column sums mu). The uniform distribution induces the famous 'Fisher-Yates' distribution and the resulting enumerative problems are of interest in applied statistics. If G=Gl(n,q) and H=K is the subgroup of upper-triangular matrices, the double cosets are indexed by permutations and the induced measure is Mallows Measure. For a continuous example, take G=O(n) (the orthogonal group) and H=K= O(n-1). The double cosets are indexed by [-1,1] and, when n=3, the induced measure is uniform. I will review work with Mackenzie Simper and Arun Ram and make sure to relate things to classical random matrix theory.

Guillaume Dubach,

• Title : Ginibre Powers and cycles of commutators

• Abstract : The eigenvalues of the complex Ginibre ensemble (matrices with i.i.d. complex Gaussian entries) form a highly correlated system of points. However, their high powers are distributed exactly as if they were independent. I will present a consequence of this counter-intuitive property to random permutations; more specifically, we will explicitly describe the distribution of the number of cycles in a commutator between a random (uniform) permutation and a fixed large cycle.

Valentin Feray,

• Title : Random partitions and random matrices: β-deformations and local limits

• Abstract : Consider a random partition (or Young diagram) distributed with Plancherel measure. It is well-known that such Young diagrams share some asymptotic properties with eigenvalues of the GUE model of random matrices: for instance, in both models, the extreme particle converges to the Tracy-Widom distribution. We will discuss here other similarities of the two models --for conditioned/fixed dimensions versions, for fluctuation of linear statistics and at the level of local limits--, as well as β-deformations.

  Based on joint work with M. Dołęga, and on a work in progress with J. Borga, C. Boutillier and P.-L. Méliot.

Franck Gabriel,

• Title : Random walks on the symmetric group as permutation invariant matrix-valued Levy processes
  

• Abstract : Continuous-time random walks on the symmetric group with uniform jumps within a given conjugacy class (e.g. transpositions) can be seen as a matrix-valued Levy process whose law is invariant by conjugation by the symmetric group. The Schur-Weyl-Jones duality asserts that the polynomial observables of these processes can be studied using the partition algebra. This approach provides natural differential equations on these observables and, consequently, allows us to explicitly compute the limiting eigenvalues distribution of the random walks when the size of the symmetric group goes to infinity. From there, we recover the critical time at which this limiting eigenvalues distribution ceases to be purely atomic, or equivalently at which the size of the largest cycle changes from microscopic to giant. 
  
  

Camille Male,

• Title :  Freeness over the diagonal and the spectrum of the sum of random matrices 

• Abstract : In this talk, I present a method to compute the spectrum of the sum of two large random matrices thanks to fixed points equations. I first review Pastur's equation and Voiculescu's subordination equations, that compute the free convolution of two probability measures and then can be applied for independent Wigner and unitarily invariant matrices. Then I introduce their generalization that has been discovered more recently to compute spectra in more general situations, for random matrices invariant in law by conjugation by permutation matrices. In collaboration with B. Au, G. Cébron, A. Dahlqvist and F. Gabriel..

Kelvin Rivera-Lopez,

• Title :  Up-down chains arising from the ordered Chinese Restaurant Process

• Abstract : In a recent paper, Leonid Petrov studied a family of Markov chains on integer partitions that arises from the Chinese Restaurant Process (CRP). These chains were shown to have a scaling limit - namely, a two-parameter family of diffusions that extends the one-parameter infinitely-many-neutral-alleles diffusions of Ethier and Kurtz. There has since been considerable interest in constructing ordered analogues of these diffusions, and it is conjectured that an ordered variant of Petrov's chains provides a construction.
  
  In this talk, I will discuss my work on this conjecture. The main objects of study will be Markov chains on integer compositions and we will analyze their convergence by obtaining triangular descriptions for their transition operators (as Petrov did). A key role will be played by the algebra of quasisymmetric functions.

  Based on joint work with Douglas Rizzolo.

Sarah Timhadjelt

• Title :  Spectral radius of the sum of a random permutation and a deterministic matrix

• Abstract : We study the asymptotic behavior of the second eigenvalue of P := S + Q where Q is a given bistochastic matrix and S is a uniformly distributed random permutation of [N]. In this goal we consider a sequence of intermediate operators Q ̃ + u independent of S which for a given N verifies that Q ̃ and Q have same distribution, u is unitary and (Q, u ̃ ) are free with amalgamation over the diagonal. Under some assumptions over the sparsity of the powers of Q we want to show that ρ(Q ̃+u) bounds asymptotically ρ(P|1⊥ ).

Harriet Walsh,

• Title : The Plancherel–Hurwitz measure: random partitions meet random maps at high genus

• Abstract : We consider a new probability measure on integer partitions, whose normalisation counts transposition factorisations on a symmetric group or equivalently discretised unconnected surfaces called Hurwitz maps. It is a deformation of the Plancherel measure, which was related to the Toda hierarchy by Okounkov and studied indirectly by Diaconis and Shahshahani in the context of random walks by transpositions. We study its asymptotic behaviour in a regime where the number of transpositions grows linearly with the order of the group and the corresponding maps are of high genus, and we find a two-fold limit shape, where the first part becomes much larger than the others. As a consequence, we obtain an asymptotic estimate for the unconnected Hurwitz numbers, which allow us to study the connectedness of a random Hurwitz map in this regime. 

  Based on joint work with Guillaume Chapuy and Baptiste Louf.

• Detailed abstract