arXiv subjects

Fabio Martinelli

Publications and source records attributed to Fabio Martinelli.

57 records · Page 4Linked to original sources

arXiv2003

Glauber dynamics on trees:Boundary conditions and mixing time

We give the first comprehensive analysis of the effect of boundary conditions on the mixing time of the Glauber dynamics in the so-called Bethe approximation. Specifically, we show that spectral gap and the log-Sobolev constant of the Glauber dynamics for the Ising model on an n-vertex regular tree with plus-boundary are bounded below by a constant independent of n at all temperatures and all external fields. This implies that the mixing time is O(log n) (in contrast to the free boundary case, where it is not bounded by any fixed polynomial at low temperatures). In addition, our methods yield simpler proofs and stronger results for the spectral gap and log-Sobolev constant in the regime where there are multiple phases but the mixing time is insensitive to the boundary condition. Our techniques also apply to a much wider class of models, including those with hard-core constraints like the antiferromagnetic Potts model at zero temperature (proper colorings) and the hard--core lattice gas (independent sets).

math.PR↗

arXiv2002

Relaxation time of anisotropic simple exclusion processes and quantum Heisenberg models

Motivated by an exact mapping between anisotropic half integer spin quantum Heisenberg models and asymmetric diffusions on the lattice, we consider an anisotropic simple exclusion process with $N$ particles in a rectangle of $\bbZ^2$. Every particle at row $h$ tries to jump to an arbitrary empty site at row $h\pm 1$ with rate $q^{\pm 1}$, where $q\in (0,1)$ is a measure of the drift driving the particles towards the bottom of the rectangle. We prove that the spectral gap of the generator is uniformly positive in $N$ and in the size of the rectangle. The proof is inspired by a recent interesting technique envisaged by E. Carlen, M.C. Carvalho and M. Loss to analyze the Kac model for the non linear Boltzmann equation. We then apply the result to prove precise upper and lower bounds on the energy gap for the spin--S, ${\rm S}\in \frac12\bbN$, XXZ chain and for the 111 interface of the spin--S XXZ Heisenberg model, thus generalizing previous results valid only for spin $\frac12$.

math.PR↗

arXiv2001

Asymmetric diffusion and the energy gap above the 111 ground state of the quantum XXZ model

We consider the anisotropic three dimensional XXZ Heisenberg ferromagnet in a cylinder with axis along the 111 direction and boundary conditions that induce ground states describing an interface orthogonal to the cylinder axis. Let $L$ be the linear size of the basis of the cylinder. Because of the breaking of the continuous symmetry around the $\hat z$ axis, the Goldstone theorem implies that the spectral gap above such ground states must tend to zero as $L\to \infty$. In \cite{BCNS} it was proved that, by perturbing in a sub--cylinder with basis of linear size $R\ll L$ the interface ground state, it is possible to construct excited states whose energy gap shrinks as $R^{-2}$. Here we prove that, uniformly in the height of the cylinder and in the location of the interface, the energy gap above the interface ground state is bounded from below by $\text{const.}L^{-2}$. We prove the result by first mapping the problem into an asymmetric simple exclusion process on $\Z^3$ and then by adapting to the latter the recursive analysis to estimate from below the spectral gap of the associated Markov generator developed in \cite{CancMart}. Along the way we improve some bounds on the equivalence of ensembles already discussed in \cite{BCNS} and we establish an upper bound on the density of states close to the bottom of the spectrum.

cond-mat.stat-mech↗