1,720,968 research outputs found
Quasi-factorization of the entropy and logarithmic Sobolev inequalities for Gibbs random fields
No full textWe 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
Cayley graphs on the symmetric group generated by initial reversals have unit spectral gap
No full textIn 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) x 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 Cayleygraph. 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
AN ALGORITHM TO STUDY TUNNELLING IN A WIDE CLASS OF ONE-DIMENSIONAL MULTIWELL POTENTIALS .1.
No full textWe 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 , where 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.
No full textWe 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
A few remarks on the octopus inequality and Aldous' spectral gap conjecture
No full textA 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
On the layering transition of an SOS surface interacting with a wall. II. The Glauber dynamics
No full textWe continue our study of the statistical mechanics of a 2D surface above a fixed wall and attracted towards it by means of a very weak positive magnetic field h in the solid on solid (SOS) approximation, when the inverse temperature beta is very large. In particular we consider a Glauber dynamics for the above model and study the rate of approach to equilibrium in a large cube with arbitrary boundary conditions. Using the results proved in the first paper of this series we show that for all h is an element of (h(k+1)*, h(k)*) ({h(k)*} being the critical values of the magnetic field found in the previous paper) the gap in the spectrum of the generator of the dynamics is bounded away from zero uniformly in the size of the box and in the boundary conditions. On the contrary, for h = h(k)* and free boundary conditions, we show that the gap in a cube of side L is bounded from above and from below by a negative exponential of L. Our results provide a strong indication that, contrary to what happens in two dimensions, for the three dimensional dynamical Ising model in a finite cube al low temperature and very small positive external field, with boundary conditions that are opposite to the field on one face of the cube and are absent (free) on the remaining faces, the rate of exponential convergence to equilibrium, which is positive in infinite volume, may go to zero exponentially fast in the side of the cube
On the eigenvalues of Cayley graphs on the symmetric group generated by a complete multipartite set of transpositions
No full textGiven a finite simple graph G with n vertices, we can construct the Cayley graph on the symmetric group S (n) generated by the edges of G, interpreted as transpositions. We show that, if G is complete multipartite, the eigenvalues of the Laplacian of Cay (G) 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 G and of Cay (G) 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
Diffusive long-time behavior of Kawasaki dynamics
No full textIn this paper we analyze the convergence to equilibrium of Kawasaki dynamics for the Ising model in the phase coexistence region. First we show, in strict analogy with the nonconservative case, that in any lattice dimension, for any boundary condition and any positive temperature and particle density, the spectral gap in a box of side L does not shrink faster than a negative exponential of the surface L^(d–1). Then we prove that, in two dimensions and for free boundary condition, the spectral gap in a box of side L is smaller than a negative exponential of L provided that the temperature is below the critical one and the particle density r satisfies r in (r+, r-), where r+, r- represent the particle density of the plus and minus phase, respectively
NONSYMMETRICAL DOUBLE WELL AND EUCLIDEAN FUNCTIONAL INTEGRAL
No full textIn this paper we show how it is possible to discuss in the language of functional integrals the problem of the symmetric double well with a small perturbation, in the semiclassical limit. This problem has been previously treated by means of a completely different approach, based on the theory of small random perturbations of dynamical systems. We recover all known results concerning the wave function and the energy splitting of the two lowest lying states, and we give an explicit expression for the prefactor of the exponential asymptotic term in the energy splitting
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