On the spectral gap of some Cayley graphs on the Weyl group W(Bn)W(B_n)

The Laplacian of a (weighted) Cayley graph on the Weyl group $W(B_n)$ is a $N\times N$ matrix with $N = 2^n n!$ equal to the order of the group. We show that for a class of (weighted) generating sets, its spectral gap (lowest nontrivial eigenvalue), is actually equal to the spectral gap of a $2n \times 2n$ matrix associated to a $2n$-dimensional permutation representation of $W_n$. This result can be viewed as an extension to $W(B_n)$ of an analogous result valid for the symmetric group, known as `Aldous' spectral gap conjecture', proven in 2010 by Caputo, Liggett and Richthammer.Comment: Version 1 (v1) contains a mistake. The main result is proved here under a less general hypothesis than in v1. Main result of v1 is left as a conjectur

Cayley graphs on the symmetric group generated by initial reversals have unit spectral gap

In a recent paper Gunnells, Scott and Walden have determined the complete spectrum of the Schreier graph on the symmetric group corresponding to the Young subgroup $S_{n-2}\times S_2$ and generated by initial reversals. In particular they find that the first nonzero eigenvalue, or spectral gap, of the Laplacian is always 1, and report that "empirical evidence" suggests that this also holds for the corresponding Cayley graph. We provide a simple proof of this last assertion, based on the decomposition of the Laplacian of Cayley graphs, into a direct sum of irreducible representation matrices of the symmetric group.Comment: Shorter version. Published in the Electron. J. of Combinatoric

On the eigenvalues of Cayley graphs on the symmetric group generated by a complete multipartite set of transpositions

Given a finite simple graph \cG with $n$ vertices, we can construct the Cayley graph on the symmetric group $S_n$ generated by the edges of \cG, interpreted as transpositions. We show that, if \cG is complete multipartite, the eigenvalues of the Laplacian of \Cay(\cG) have a simple expression in terms of the irreducible characters of transpositions, and of the Littlewood-Richardson coefficients. As a consequence we can prove that the Laplacians of \cG and of \Cay(\cG) have the same first nontrivial eigenvalue. This is equivalent to saying that Aldous's conjecture, asserting that the random walk and the interchange process have the same spectral gap, holds for complete multipartite graphs.Comment: 29 pages. Includes modification which appear on the published version in J. Algebraic Combi

D-COMPONENT ROTATORS AS THE CLASSICAL LIMIT OF QUANTUM SO(D) VECTOR MODELS

The author considers a class of spin systems whose single-site configuration space is an orbit of a representation of a compact Lie group G. For these models the author gets upper and lower bounds to the quantum partition function in terms of two classical partition functions. If a certain group-theoretic condition is satisfied, then these bounds allow one to prove the convergence of a suitable sequence of quantum partition functions to the 'corresponding' classical one. This condition is shown to be satisfied, in particular, for the D-component rotators when D is odd. The result could be useful for the extension of the Lee-Yang theorem to such models

AN ALGORITHM TO STUDY TUNNELLING IN A WIDE CLASS OF ONE-DIMENSIONAL MULTIWELL POTENTIALS .1.

We prove a theorem which gives an algorithmic solution to the problem of finding the logarithmic derivative of the ground state wave function of one dimensional systems. By means of this quantity, as it is well known, one can determine the lowest part of the spectrum of the Hamiltonian by probabilistic methods. We show that, in some natural classes of potentials, the complexity of our algorithm is less than $N^3$, where $N$ is the number of the absolute minima of the potential. Our approach allows a systematic treatment of cases of much higher complexity than those analyzed so far in the literature and it can be useful in the study of physical systems like, for example, long molecular chains or superlattice structures

AN ALGORITHM TO STUDY TUNNELING IN A WIDE CLASS OF ONE-DIMENSIONAL MULTIWELL POTENTIALS .2.

We describe a simple algorithm to determine the behaviour of the ground state wave function and to compute the lowest part of the spectrum of the Schroedinger operator in the semiclassical limit, when the potential has many absolute minima. This approach may be useful in the study of complex systems, like long molecular chains or superlattice structures. As an application we determine the number of the energy levels in the first band if the potential is a binary sequence of two types of barriers, and give a method to handle more general cases. We estimate statistically the versatility of our algorithm, with the aid of a computer program that implements it. It turns out that 99% of potentials in some general classes are solvable with our method

Quasi-factorization of the entropy and logarithmic Sobolev inequalities for Gibbs random fields

We show that the entropy functional exhibits a quasi-factorization property with respect to a pair of weakly dependent sigma -algebras. As an application we give a simple proof that the Dobrushin and Shlosman's complete analyticity condition, for a Gibbs specification with finite range summable interaction, implies uniform logarithmic Sobolev inequalities. This result has been previously proven using several different techniques. The advantage of our approach is that it relies almost entirely on a general property of the entropy, while very little is assumed on the Dirichlet form. No topology is introduced on the single spin space, thus discrete and continuous spins can be treated in the same way

A few remarks on the octopus inequality and Aldous' spectral gap conjecture

A conjecture by D. Aldous, which can be formulated as a statement about the first nontrivial eigenvalue of the Laplacian of certain Cayley graphs on the symmetric group generated by transpositions, has been recently proven by Caputo, Liggett, and Richthammer. Their proof is a subtle combination of two ingredients: a nonlinear mapping in the group algebra which permits a proof by induction, and a quite hard estimate named the octopus inequality. In this article we present a simpler and more transparent proof of the octopus inequality, which emerges naturally when looking at the Aldous’ conjecture from an algebraic perspective