484 research outputs found
Mod-Gaussian convergence and its applications for models of statistical mechanics
In this paper we complete our understanding of the role played by the
limiting (or residue) function in the context of mod-Gaussian convergence. The
question about the probabilistic interpretation of such functions was initially
raised by Marc Yor. After recalling our recent result which interprets the
limiting function as a measure of "breaking of symmetry" in the Gaussian
approximation in the framework of general central limit theorems type results,
we introduce the framework of -mod-Gaussian convergence in which the
residue function is obtained as (up to a normalizing factor) the probability
density of some sequences of random variables converging in law after a change
of probability measure. In particular we recover some celebrated results due to
Ellis and Newman on the convergence in law of dependent random variables
arising in statistical mechanics. We complete our results by giving an
alternative approach to the Stein method to obtain the rate of convergence in
the Ellis-Newman convergence theorem and by proving a new local limit theorem.
More generally we illustrate our results with simple models from statistical
mechanics.Comment: 49 pages, 21 figure
Identities from representation theory
We give a new Jacobi--Trudi-type formula for characters of finite-dimensional
irreducible representations in type using characters of the fundamental
representations and non-intersecting lattice paths. We give equivalent
determinant formulas for the decomposition multiplicities for tensor powers of
the spin representation in type and the exterior representation in type
. This gives a combinatorial proof of an identity of Katz and equates such
a multiplicity with the dimension of an irreducible representation in type
. By taking certain specializations, we obtain identities for -Catalan
triangle numbers, the -Catalan number of Stump, -triangle versions of
Motzkin and Riordan numbers, and generalizations of Touchard's identity. We use
(spin) rigid tableaux and crystal base theory to show some formulas relating
Catalan, Motzkin, and Riordan triangle numbers.Comment: 68 pages, 8 figure
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