Valentin Féray

Cette thèse concerne les valeurs du caractère irréductible (renormalisé) comme fonction de la partition indexant la représentation (et non de la permutation sur laquelle on calcule le caractère). Avec une bonne renormalisation, les caractères s’écrivent comme des polynômes en fonction des coordonnées des diagrammes multirectangulaires d’une part et en fonction des cumulants libres d’autre part ( ce sont des observables du diagramme apparaissant naturellement dans des problèmes d’asymptotique). Nous avons donné des interprétations combinatoires des coefficients de ces différentes expressions. Celles-ci peuvent s’exprimer en termes de cartes, dont le genre est lié au comportement asymptotique du terme correspondant. Ce type d’expression permet d’une part de bien comprendre le comportement asymptotique : nous avons ainsi amélioré les bornes connues sur les caractères ainsi que le domaine de validité d’équivalents classique. D’autre part, la combinatoire apparaissant dans ces questions est riche et a pu être utilisée dans l’étude d’identité sur des fractions rationnelles

We consider uniform random permutations in classes having a finite combinatorial specification for the substitution decomposition. These classes include (but are not limited to) all permutation classes with a finite number of simple permutations. Our goal is to study their limiting behavior in the sense of permutons. The limit depends on the structure of the specification restricted to families with the largest growth rate. When it is strongly connected, two cases occur. If the associated system of equations is linear, the limiting permuton is a deterministic $X$-shape. Otherwise, the limiting permuton is the Brownian separable permuton, a random object that already appeared as the limit of most substitution-closed permutation classes, among which the separable permutations. Moreover these results can be combined to study some non strongly connected cases. To prove our result, we use a characterization of the convergence of random permutons by the convergence of random subpermutations. Key steps are the combinatorial study, via substitution trees, of families of permutations with marked elements inducing a given pattern, and the singularity analysis of the corresponding generating functions.

Weighted dependency graphs have been recently introduced by the second author, as a toolbox to prove central limit theorems. In this paper, we prove that spins in the d -dimensional Ising model display such a weighted dependency structure. We use this to obtain various central limit theorems for the number of occurrences of local and global patterns in a growing box.

We use the recently developed method of weighted dependency graphs to prove central limit theorems for the number of occurrences of any fixed pattern in multiset permutations and in set partitions. This generalizes results for patterns of size 2 in both settings, obtained by Canfield, Janson and Zeilberger and Chern, Diaconis, Kane and Rhoades, respectively.

We are interested in the asymptotics of the number of standard Young tableaux f λ / μ f^{\lambda /\mu } of a given skew shape λ / μ \lambda /\mu . We mainly restrict ourselves to the case where both diagrams are balanced, but investigate all growth regimes of | μ | |\mu | compared to | λ | |\lambda | , from | μ | |\mu | fixed to | μ | |\mu | of order | λ | |\lambda | . When | μ | = o ( | λ | 1 / 3 ) |\mu |=o(|\lambda |^{1/3}) , we get an asymptotic expansion to any order. When | μ | = o ( | λ | 1 / 2 ) |\mu |=o(|\lambda |^{1/2}) , we get a sharp upper bound. For larger | μ | |\mu | , we prove a weaker bound and give a conjecture on what we believe to be the correct order of magnitude. Our results are obtained by expressing f λ / μ f^{\lambda /\mu } in terms of irreducible character values of the symmetric group and applying known upper bounds on characters.