58 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
The spectrum of local random Hamiltonians
The spectrum of a local random Hamiltonian can be represented generically by
the so-called -free convolution of its local terms' probability
distributions. We establish an isomorphism between the set of
-noncrossing partitions and permutations to study its spectrum.
Moreover, we derive some lower and upper bounds for the largest eigenvalue of
the Hamiltonian.Comment: 22 page
Recommended from our members
On the homotopy type of multipath complexes
A multipath in a directed graph is a disjoint union of paths. The multipath complex of a directed graph G is the simplicial complex whose faces are the multipaths of G. We compute Euler characteristics, and associated generating functions, of the multipath complexes of directed graphs from certain families, including transitive tournaments and complete bipartite graphs. We show that if G is a linear graph, polygon, small grid or transitive tournament, then the homotopy type of the multipath complex of G is always contractible or a wedge of spheres. We introduce a new technique for decomposing directed graphs into dynamical regions, which allows us to simplify the homotopy computations
On the E-polynomial of a familiy of parabolic Sp2n-character varieties
In this thesis, we find the E-polynomials of a family of parabolic symplectic character
varieties of Riemann surfaces by constructing a stratification, proving that
each stratum has polynomial count, applying a result of Katz regarding the
counting functions, and finally adding up the resulting E-polynomials of the
strata. To count the number of rational points of the strata, we invoke a formula
due to Frobenius. Our calculation make use of a formula for the evaluation of
characters on semisimple elements coming from Deligne-Lusztig theory, applied
to the character theory of the finite symplectic group, and Möbius inversion on the poset of
set-partitions. We compute the Euler characteristic of the our character varieties with these
polynomials, and show they are connected
Recombination models forward and backward in time
Esser M. Recombination models forward and backward in time. Bielefeld: Universität Bielefeld; 2017
On sensitivity in bipartite Cayley graphs
Huang proved that every set of more than half the vertices of the
-dimensional hypercube induces a subgraph of maximum degree at least
, which is tight by a result of Chung, F\"uredi, Graham, and Seymour.
Huang asked whether similar results can be obtained for other highly symmetric
graphs.
First, we present three infinite families of Cayley graphs of unbounded
degree that contain induced subgraphs of maximum degree on more than half
the vertices. In particular, this refutes a conjecture of Potechin and Tsang,
for which first counterexamples were shown recently by Lehner and Verret. The
first family consists of dihedrants and contains a sporadic counterexample
encountered earlier by Lehner and Verret. The second family are star graphs,
these are edge-transitive Cayley graphs of the symmetric group. All members of
the third family are -regular containing an induced matching on a
-fraction of the vertices. This is largest possible and answers
a question of Lehner and Verret.
Second, we consider Huang's lower bound for graphs with subcubes and show
that the corresponding lower bound is tight for products of Coxeter groups of
type , , and most exceptional cases. We
believe that Coxeter groups are a suitable generalization of the hypercube with
respect to Huang's question.
Finally, we show that induced subgraphs on more than half the vertices of
Levi graphs of projective planes and of the Ramanujan graphs of Lubotzky,
Phillips, and Sarnak have unbounded degree. This gives classes of Cayley graphs
with properties similar to the ones provided by Huang's results. However, in
contrast to Coxeter groups these graphs have no subcubes.Comment: 20 pages, 4 figures, 2 tables, improved section
Morita equivalence and decomposition spaces
Inspired by prior results by Stanley and Leroux showing what information can be recovered from an isomorphism of incidence algebras, we investigate the very same idea applied to decomposition spaces. We review the work of Stanley and Leroux and provide sufficient background on the homotopy theory of groupoids to be able to define decomposition spaces, equivalences of them as linear functors and solve the isomorphism problem for both the groupoid-level coalgebra and for the numerical incidence algebra
- …