52 research outputs found
Absence of Dobrushin states for long-range Ising models
We consider the two-dimensional Ising model with long-range pair interactions
of the form with , mostly when . We show that Dobrushin states (i.e. extremal non-translation-invariant
Gibbs states selected by mixed -boundary conditions) do not exist. We
discuss possible extensions of this result in the direction of the
Aizenman-Higuchi theorem, or concerning fluctuations of interfaces. We also
mention the existence of rigid interfaces in two long-range anisotropic
contexts.Comment: revised versio
Gibbs-non-Gibbs properties for n-vector lattice and mean-field models
We review some recent developments in the study of Gibbs and non-Gibbs
properties of transformed n-vector lattice and mean-field models under various
transformations. Also, some new results for the loss and recovery of the Gibbs
property of planar rotor models during stochastic time evolution are presented.Comment: 31 pages, 6 figure
Local central limit theorem and potential kernel estimates for a class of symmetric heavy-tailted random variables
In this article, we study a class of heavy-tailed random variables on
in the domain of attraction of an -stable random variable
of index satisfying a certain expansion of their
characteristic function. Our results include sharp convergence rates for the
local (stable) central limit theorem of order , a
detailed expansion of the characteristic function of a long-range random walk
with transition probability proportional to and and furthermore detailed asymptotic estimates of the discrete potential
kernel (Green's function) up to order for any small
enough, when .Comment: 33 page
Contour methods for long-range Ising models: weakening nearest-neighbor interactions and adding decaying fields
We consider ferromagnetic long-range Ising models which display phase
transitions. They are long-range one-dimensional Ising ferromagnets, in which
the interaction is given by with , in particular, .
For this class of models one way in which one can prove the phase transition is
via a kind of Peierls contour argument, using the adaptation of the
Fr\"ohlich-Spencer contours for , proposed by Cassandro,
Ferrari, Merola and Presutti. As proved by Fr\"ohlich and Spencer for
and conjectured by Cassandro et al for the region they could treat,
for , although in the
literature dealing with contour methods for these models it is generally
assumed that , we can show that this condition can be removed in the
contour analysis. In addition, combining our theorem with a recent result of
Littin and Picco we prove the persistence of the contour proof of the phase
transition for any . Moreover, we show that when we add a
magnetic field decaying to zero, given by and
where , the
transition still persists.Comment: 13 page
Scaling limit of the odometer in divisible sandpiles
In a recent work [LMPU] prove that the odometer function of a divisible sandpile model on a finite graph can be expressed as a shifted discrete bilaplacian Gaussian field. For the discrete torus, they suggest the possibility that the scaling limit of the odometer may be related to the continuum bilaplacian field. In this work we show that in any dimension the rescaled odometer converges to the continuum bilaplacian field on the unit torus
The divisible sandpile with heavy-tailed variables
This work deals with the divisible sandpile model when an initial configuration sampled from a heavy-tailed distribution. Extending results of Levine et al. (2015) and Cipriani et al. (2016) we determine sufficient conditions for stabilization and non-stabilization on infinite graphs. We determine furthermore that the scaling limit of the odometer on the torus is an α-stable random distribution
Gibbsianness versus Non-Gibbsianness of time-evolved planar rotor models
We study the Gibbsian character of time-evolved planar rotor systems on Z^d,
d at least 2, in the transient regime, evolving with stochastic dynamics and
starting with an initial Gibbs measure. We model the system by interacting
Brownian diffusions, moving on circles. We prove that for small times and
arbitrary initial Gibbs measures \nu, or for long times and both high- or
infinite-temperature measure and dynamics, the evolved measure \nu^t stays
Gibbsian. Furthermore we show that for a low-temperature initial measure \nu,
evolving under infinite-temperature dynamics thee is a time interval (t_0, t_1)
such that \nu^t fails to be Gibbsian in d=2.Comment: latexpdf, with 2 pdf figure
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