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 $\Z^2$ 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 $\mathbb{Z}^2$. 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