78 research outputs found
Range of a Transient 2d-Random Walk
We study the range of a planar random walk on a randomly oriented lattice,
already known to be transient. We prove that the expectation of the range grows
linearly, in both the quenched (for a.e. orientation) and annealed ("averaged")
cases. We also express the rate of growth in terms of the quenched Green
function and eventually prove a weak law of large numbers in the
(non-Markovian) annealed case.Comment: 9 page
Parsimonious Description of Generalized Gibbs Measures : Decimation of the 2d-Ising Model
In this paper, we detail and complete the existing characterizations of the
decimation of the Ising model on in the generalized Gibbs context. We
first recall a few features of the Dobrushin program of restoration of
Gibbsianness and present the construction of global specifications consistent
with the extremal decimated measures. We use them to consider these
renormalized measures as almost Gibbsian measures and to precise its convex set
of DLR measures. We also recall the weakly Gibbsian description and complete it
using a potential that admits a quenched correlation decay, i.e. a well-defined
configuration-dependent length beyond which this potential decays
exponentially. We use these results to incorporate these decimated measures in
the new framework of parsimonious random fields that has been recently
developed to investigate probability aspects related to neurosciences.Comment: 32 pages, preliminary versio
Spin-Flip Dynamics of the Curie-Weiss Model: Loss of Gibbsianness with Possibly Broken Symmetry
We study the conditional probabilities of the Curie-Weiss Ising model in vanishing external field under a symmetric independent stochastic spin-flip dynamics and discuss their set of points of discontinuity (bad points). We exhibit a complete analysis of the transition between Gibbsian and non-Gibbsian behavior as a function of time, extending the results for the corresponding lattice model, where only partial answers can be obtained. For initial temperature β^−1 ≥ 1, we prove that the time-evolved measure is always Gibbsian. For ⅔ ≤ β^−1 < 1, the time-evolved measure loses its Gibbsian character at a sharp transition time. For β^−1 < ⅔, we observe the new phenomenon of symmetry-breaking in the set of points of discontinuity: Bad points corresponding to non-zero spin-average appear at a sharp transition time and give rise to biased non-Gibbsianness of the time-evolved measure. These bad points become neutral at a later transition time, while the measure stays non-Gibbs. In our proof we give a detailed description of the phase-diagram of a Curie-Weiss random field Ising model with possibly non-symmetric random field distribution based on bifurcation analysis.
A functional limit theorem for a 2D-random walk with dependent marginals
We prove a non-standard functional limit theorem for a two dimensional simple
random walk on some randomly oriented lattices. This random walk, already known
to be transient, has different horizontal and vertical fluctuations leading to
different normalizations in the functional limit theorem, with a non-Gaussian
horizontal behavior. We also prove that the horizontal and vertical components
are not asymptotically independent
Random walks on FKG-horizontally oriented lattices
We study the asymptotic behavior of the simple random walk on oriented
version of . The considered latticesare not directed on the
vertical axis but unidirectional on the horizontal one, with symmetric random
orientations which are positively correlated. We prove that the simple random
walk is transient and also prove a functionnal limit theorem in the space of
cadlag functions, with an unconventional normalization.Comment: 16 page
Decimation of the Dyson-Ising Ferromagnet
We study the decimation to a sublattice of half the sites, of the
one-dimensional Dyson-Ising ferromagnet with slowly decaying long-range pair
interactions of the form , in the phase transition
region (1< 2, and low temperature). We prove non-Gibbsianness of
the decimated measure at low enough temperatures by exhibiting a point of
essential discontinuity for the finite-volume conditional probabilities of
decimated Gibbs measures. Thus result complements previous work proving
conservation of Gibbsianness for fastly decaying potentials ( > 2) and
provides an example of a "standard" non-Gibbsian result in one dimension, in
the vein of similar resuts in higher dimensions for short-range models. We also
discuss how these measures could fit within a generalized (almost vs. weak)
Gibbsian framework. Moreover we comment on the possibility of similar results
for some other transformations.Comment: 18 pages, some corrections and references added, to appear in
Stoch.Proc.App
Gibbs Measures for Long-Range Ising Models
This review-type paper is based on a talk given at the conference États de la Recherche en Mécanique statistique, which took place at IHP in Paris (December 10-14, 2018). We revisit old results from the 80's about one dimensional long-range polynomially decaying Ising models (often called Dyson models in dimension one) and describe more recent results about interface fluctuations and interface states in dimensions one and two
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