Random walks in a random environment on a strip: a renormalization group approach

We present a real space renormalization group scheme for the problem of random walks in a random environment on a strip, which includes one-dimensional random walk in random environment with bounded non-nearest-neighbor jumps. We show that the model renormalizes to an effective one-dimensional random walk problem with nearest-neighbor jumps and conclude that Sinai scaling is valid in the recurrent case, while in the sub-linear transient phase, the displacement grows as a power of the time.Comment: 9 page

Poisson smooth structures on stratified symplectic spaces

In this paper we introduce the notion of a smooth structure on a stratified space, the notion of a Poisson smooth structure and the notion of a weakly symplectic smooth structure on a stratified symplectic space, refining the concept of a stratified symplectic Poisson algebra introduced by Sjamaar and Lerman. We show that these smooth spaces possess several important properties, e.g. the existence of smooth partitions of unity. Furthermore, under mild conditions many properties of a symplectic manifold can be extended to a symplectic stratified space provided with a smooth Poisson structure, e.g. the existence and uniqueness of a Hamiltonian flow, the isomorphism between the Brylinski-Poisson homology and the de Rham homology, the existence of a Leftschetz decomposition on a symplectic stratified space. We give many examples of stratified symplectic spaces possessing a Poisson smooth structure which is also weakly symplectic.Comment: 21 page, final version, to appear in the Proceedings of the 6-th World Conference on 21st Century Mathematic

On the zero-temperature limit of Gibbs states

We exhibit Lipschitz (and hence H\"older) potentials on the full shift $\{0,1\}^{\mathbb{N}}$ such that the associated Gibbs measures fail to converge as the temperature goes to zero. Thus there are "exponentially decaying" interactions on the configuration space $\{0,1\}^{\mathbb Z}$ for which the zero-temperature limit of the associated Gibbs measures does not exist. In higher dimension, namely on the configuration space $\{0,1\}^{\mathbb{Z}^{d}}$, $d\geq3$, we show that this non-convergence behavior can occur for finite-range interactions, that is, for locally constant potentials.Comment: The statement of Theorem 1.2 is more accurate and some new comment follow i

Lingering random walks in random environment on a strip

We consider a recurrent random walk (RW) in random environment (RE) on a strip. We prove that if the RE is i. i. d. and its distribution is not supported by an algebraic subsurface in the space of parameters defining the RE then the RW exhibits the "(log t)-squared" asymptotic behaviour. The exceptional algebraic subsurface is described by an explicit system of algebraic equations. One-dimensional walks with bounded jumps in a RE are treated as a particular case of the strip model. If the one dimensional RE is i. i. d., then our approach leads to a complete and constructive classification of possible types of asymptotic behaviour of recurrent random walks. Namely, the RW exhibits the $(\log t)^{2}$ asymptotic behaviour if the distribution of the RE is not supported by a hyperplane in the space of parameters which shall be explicitly described. And if the support of the RE belongs to this hyperplane then the corresponding RW is a martingale and its asymptotic behaviour is governed by the Central Limit Theorem

Chaotic temperature dependence at zero temperature

We present a class of examples of nearest-neighbour, boubded-spin models, in which the low-temperature Gibbs measures do not converge as the temperature is lowered to zero, in any dimension

The Analyticity of a Generalized Ruelle's Operator

In this work we propose a generalization of the concept of Ruelle operator for one dimensional lattices used in thermodynamic formalism and ergodic optimization, which we call generalized Ruelle operator, that generalizes both the Ruelle operator proposed in [BCLMS] and the Perron Frobenius operator defined in [Bowen]. We suppose the alphabet is given by a compact metric space, and consider a general a-priori measure to define the operator. We also consider the case where the set of symbols that can follow a given symbol of the alphabet depends on such symbol, which is an extension of the original concept of transition matrices from the theory of subshifts of finite type. We prove the analyticity of the Ruelle operator and present some examples

Flatness is a Criterion for Selection of Maximizing Measures

For a full shift with Np+1 symbols and for a non-positive potential, locally proportional to the distance to one of N disjoint full shifts with p symbols, we prove that the equilibrium state converges as the temperature goes to 0. The main result is that the limit is a convex combination of the two ergodic measures with maximal entropy among maximizing measures and whose supports are the two shifts where the potential is the flattest. In particular, this is a hint to solve the open problem of selection, and this indicates that flatness is probably a/the criterion for selection as it was conjectured by A.O. Lopes. As a by product we get convergence of the eigenfunction at the log-scale to a unique calibrated subaction