arXiv subjects

F. Redig

Publications and source records attributed to F. Redig.

32 records · Page 2Linked to original sources

arXiv2006

Concentration inequalities for random fields via coupling

We present a new and simple approach to concentration inequalities for functions around their expectation with respect to non-product measures, i.e., for dependent random variables. Our method is based on coupling ideas and does not use information inequalities. When one has a uniform control on the coupling, this leads to exponential concentration inequalities. When such a uniform control is no more possible, this leads to polynomial or stretched-exponential concentration inequalities. Our abstract results apply to Gibbs random fields, in particular to the low-temperature Ising model which is a concrete example of non-uniformity of the coupling.

math.PR↗

arXiv2005

Freezing transitions in non-Fellerian particle systems

Non-Fellerian processes show phenomena that are unseen in standard interacting particle systems. We consider freezing transitions in one-dimensional non-Fellerian processes which are built from the abelian sandpile additions to which in one case, spin flips are added, and in another case, the so called anti-sandpile subtractions. In the first case and as a function of the sandpile addition rate, there is a sharp transition from a non-trivial invariant measure to the invariant measure of the sandpile process. For the combination sandpile plus anti-sandpile, there is a sharp transition from one frozen state to the other anti-state

math.PR↗

arXiv2005

Organized versus self-organized criticality in the abelian sandpile model

We define stabilizability of an infinite volume height configuration and of a probability measure on height configurations. We show that for high enough densities, a probability measure cannot be stabilized. We also show that in some sense the thermodynamic limit of the uniform measures on the recurrent configurations of the abelian sandpile model (ASM) is a maximal element of the set of stabilizable measures. In that sense the self-organized critical behavior of the ASM can be understood in terms of an ordinary transition between stabilizable and non-stabilizable

math-ph↗

arXiv2005

Testing the irreversibility of a Gibbsian process via hitting and return times

We introduce estimators for the entropy production of a Gibbsian process based on the observation of a single or two typical trajectories. These estimators are built with adequate hitting and return times. We then study their convergence and fluctuation properties. This provides statisticals test for the irreversibility of Gibbsian processes.

math-ph↗

arXiv2005

Occurrence, repetition and matching of patterns in the low-temperature Ising model

We continue our study of exponential law for occurrences and returns of patterns in the context of Gibbsian random fields. For the low temperature plus phase of the Ising model, we prove exponential laws with error bounds for occurrence, return, waiting and matching times. Moreover we obtain a Poisson law for the number of occurrences of large cylindrical events and a Gumbel law for the maximal overlap between two independent copies. As a by-product, we derive precise fluctuation results for the logarithm of waiting and return times. The main technical tool we use, in order to control mixing, is disagreement percolation.

math-ph↗

arXiv2004

Loss without recovery of Gibbsianness during diffusion of continuous spins

We consider a specific continuous-spin Gibbs distribution $μ_{t=0}$ for a double-well potential that allows for ferromagnetic ordering. We study the time-evolution of this initial measure under independent diffusions. For `high temperature' initial measures we prove that the time-evoved measure $μ_{t}$ is Gibbsian for all $t$. For `low temperature' initial measures we prove that $μ_t$ stays Gibbsian for small enough times $t$, but loses its Gibbsian character for large enough $t$. In contrast to the analogous situation for discrete-spin Gibbs measures, there is no recovery of the Gibbs property for large $t$ in the presence of a non-vanishing external magnetic field. All of our results hold for any dimension $d\geq 2$. This example suggests more generally that time-evolved continuous-spin models tend to be non-Gibbsian more easily than their discrete-spin counterparts.

math-ph↗

arXiv2004

Large deviations for quantum spin systems

We consider high temperature KMS states for quantum spin systems on a lattice. We prove a large deviation principle for the distribution of empirical averages $\frac{1}{|Λ|} \sum_{i\inΛ} X_i$, where the $X_i$'s are copies of a self-adjoint element $X$ (level one large deviations). From the analyticity of the generating function, we obtain the central limit theorem. We generalize to a level two large deviation principle for the distribution of $\frac{1}{|Λ|}\sum_{i\inΛ} δ_{X_i}$.

math-ph↗

arXiv2003

The Potts model built on sand

We consider the q=4 Potts model on the square lattice with an additional hard-core nonlocal interaction. That interaction arises from the choice of the reference measure taken to be the uniform measure on the recurrent configurations for the abelian sandpile model. In that reference measure some correlation functions have a power-law decay. We investigate the low-temperature phase diagram and we prove the existence of a single stable phase with exponential decay of correlations. For all boundary conditions the density of 4 in the infinite volume limit goes to one as the temperature tends to zero.

cond-mat.stat-mech↗

arXiv2003

Exponential distribution for the occurrence of rare patterns in Gibbsian random fields

We study the distribution of the occurrence of rare patterns in sufficiently mixing Gibbs random fields on the lattice $\mathbb{Z}^d$, $d\geq 2$. A typical example is the high temperature Ising model. This distribution is shown to converge to an exponential law as the size of the pattern diverges. Our analysis not only provides this convergence but also establishes a precise estimate of the distance between the exponential law and the distribution of the occurrence of finite patterns. A similar result holds for the repetition of a rare pattern. We apply these results to the fluctuation properties of occurrence and repetition of patterns: We prove a central limit theorem and a large deviation principle.

math.PR↗

arXiv2003

The Abelian sandpile; a mathematical introduction

We give a simple rigourous treatment of the classical results of the abelian sandpile model. Although we treat results which are well-known in the physics literature, in many cases we did not find complete proofs in the literature. The paper tries to fill the gap between the mathematics and the physics literature on this subject, and also presents some new proofs. It can also serve as an introduction to the model.

cond-mat.soft↗

arXiv2001

No current without heat

We show for a large class of interacting particle systems that whenever the stationary measure is not reversible for the dynamics, then the mean entropy production in the steady state is strictly positive. This extends to the thermodynamic limit the equivalence between microscopic reversibility and zero mean entropy production: time-reversal invariance cannot be spontaneously broken.

cond-mat.stat-mech↗

arXiv2001

Possible loss and recovery of Gibbsianness during the stochastic evolution of Gibbs measures

We consider Ising-spin systems starting from an initial Gibbs measure $ν$ and evolving under a spin-flip dynamics towards a reversible Gibbs measure $μ\not=ν$. Both $ν$ and $μ$ are assumed to have a finite-range interaction. We study the Gibbsian character of the measure $νS(t)$ at time $t$ and show the following: (1) For all $ν$ and $μ$, $νS(t)$ is Gibbs for small $t$. (2) If both $ν$ and $μ$ have a high or infinite temperature, then $νS(t)$ is Gibbs for all $t>0$. (3) If $ν$ has a low non-zero temperature and a zero magnetic field and $μ$ has a high or infinite temperature, then $νS(t)$ is Gibbs for small $t$ and non-Gibbs for large $t$. (4) If $ν$ has a low non-zero temperature and a non-zero magnetic field and $μ$ has a high or infinite temperature, then $νS(t)$ is Gibbs for small $t$, non-Gibbs for intermediate $t$, and Gibbs for large $t$. The regime where $μ$ has a low or zero temperature and $t$ is not small remains open. This regime presumably allows for many different scenarios.

math-ph↗

arXiv1998

On the thermodynamic limit for a one-dimensional sandpile process

Considering the standard abelian sandpile model in one dimension, we construct an infinite volume Markov process corresponding to its thermodynamic (infinite volume) limit. The main difficulty we overcome is the strong non-locality of the dynamics. However, using similar ideas as in recent extensions of the standard Gibbs formalism for lattice spin systems, we can identify a set of `good' configurations on which the dynamics is effectively local. We prove that every configuration converges in a finite time to the unique invariant measure.

math.PR↗

arXiv1998

The restriction of the Ising model to a layer

We discuss the status of recent Gibbsian descriptions of the restriction (projection) of the Ising phases to a layer. We concentrate on the projection of the two-dimensional low temperature Ising phases for which we prove a variational principle.

math.PR↗