Convergence to equilibrium for finite Markov processes, with application to the Random Energy Model

We estimate the distance in total variation between the law of a finite state Markov process at time t, starting from a given initial measure, and its unique invariant measure. We derive upper bounds for the time to reach the equilibrium. As an example of application we consider a special case of finite state Markov process in random environment: the Metropolis dynamics of the Random Energy Model. We also study the process of the environment as seen from the process

Carne--Varopoulos bounds for centered random walks

We extend the Carne--Varopoulos upper bound on the probability transitions of a Markov chain to a certain class of nonreversible processes by introducing the definition of a ``centering measure.'' In the case of random walks on a group, we study the connections between different notions of centering.Comment: Published at http://dx.doi.org/10.1214/009117906000000052 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Isoperimetry and heat kernel decay on percolation clusters

We prove that the heat kernel on the infinite Bernoulli percolation cluster in Z^d almost surely decays faster than t^{-d/2}. We also derive estimates on the mixing time for the random walk confined to a finite box. Our approach is based on local isoperimetric inequalities

On the dynamics of trap models in Z^d

We consider trap models on Z^d, namely continuous time Markov jump process on Z^d with embedded chain given by a generic discrete time random walk, and whose mean waiting time at x is given by tau_x, with tau = (tau_x, x in Z^d) a family of positive iid random variables in the basin of attraction of an alpha-stable law, 0<alpha<1. We may think of x as a trap, and tau_x as the depth of the trap at x. We are interested in the trap process, namely the process that associates to time t the depth of the currently visited trap. Our first result is the convergence of the law of that process under suitable scaling. The limit process is given by the jumps of a certain alpha-stable subordinator at the inverse of another alpha-stable subordinator, correlated with the first subordinator. For that result, the requirements for the embedded random walk are a) the validity of a law of large numbers for its range, and b) the slow variation at infinity of the tail of the distribution of its time of return to the origin: they include all transient random walks as well as all planar random walks, and also many one dimensional random walks. We then derive aging results for the process, namely scaling limits for some two-time correlation functions thereof, a strong form of which requires an assumption of transience, stronger than a, b. The above mentioned scaling limit result is an averaged result with respect to tau. Under an additional condition on the size of the intersection of the ranges of two independent copies of the embeddded random walk, roughly saying that it is small compared with the size of the range, we derive a stronger scaling limit result, roughly stating that it holds in probability with respect to tau. With that additional condition, we also strengthen the aging results, from the averaged version mentioned above, to convergence in probability with respect to tau.Comment: 36 pages, 5 figures. Replaces first version, with a correction to/weakening of the statement of current Theorem 25, corrections to its proof and that of Lemma 16, the addition of a subsection on integrated aging results. Section 4, on convergence, was somewhat restructure

Co-reduction of aluminium and lanthanide ions in molten fluorides : application to cerium and samarium extraction from nuclear waste

This work concerns the method of co-reduction process with aluminium ions in LiF–CaF2 medium (79–21 mol.%) on tungsten electrode for cerium and samarium extraction. Electrochemical techniques such as cyclic and square wave voltammetries, and potentiostatic electrolyses were used to study the co-reduction of CeF3 and SmF3 with AlF3. For each of these elements, specific peaks of Al–Ce and Al–Sm alloys formationwere observed by voltammetry aswell as peaks of pure cerium and aluminium, and pure samarium and aluminium respectively. The difference of potential measured between the solvent reduction and the alloy formation suggests expecting an extraction efficiency of 99.99% of each lanthanide by the process. Different intermetallic compounds were obtained for different potentiostatic electrolysis and were characterised by Scanning Electron Microscopy with EDS probe. The validity of the process was verified by carrying out cerium and samarium extractions in the form of Al–Ln alloy; the extraction efficiency was 99.5% for Ce(III) and 99.4% for Sm(III)

Schur Superpolynomials: Combinatorial Definition and Pieri Rule

Schur superpolynomials have been introduced recently as limiting cases of the Macdonald superpolynomials. It turns out that there are two natural super-extensions of the Schur polynomials: in the limit $q=t=0$ and $q=t\rightarrow\infty$, corresponding respectively to the Schur superpolynomials and their dual. However, a direct definition is missing. Here, we present a conjectural combinatorial definition for both of them, each being formulated in terms of a distinct extension of semi-standard tableaux. These two formulations are linked by another conjectural result, the Pieri rule for the Schur superpolynomials. Indeed, and this is an interesting novelty of the super case, the successive insertions of rows governed by this Pieri rule do not generate the tableaux underlying the Schur superpolynomials combinatorial construction, but rather those pertaining to their dual versions. As an aside, we present various extensions of the Schur bilinear identity

Path representation of su(2)_k states I: Operators and particles for k=1,2

This is the first of two articles devoted to the analysis of the path description of the states in su(2)_k WZW models, a representation well suited for constructive derivations of the fermionic characters. In this first article, the cases k=1,2 are treated in detail, emphasizing a different description in each case (operators vs particles). For k=1, we first prove, as a side result, the equivalence of two known path representations for the finitized su(2)_1 states by displaying an explicit bijection. An immediate offshoot is the gain of a new and simple weighting for the (Kyoto) path representation that generalizes to level k. The bijection also suggests two operator constructions for the su(2)_1 paths, a local and a nonlocal one, both interrelated. These are formal operators that map a path to another path, so that any path can be obtained by successive applications of these operators on a simple reference (ground-state) path. The nonlocal operator description is the starting point for a direct and elementary derivation of the su(2)_1 spinon character. The second part presents an extensive study of the su(2)_2 paths from their particle point of view, where the particles are defined as the path building blocks. The resulting generating functions appear to provide new (at least superficially) fermionic forms of the characters. In particular, a nice relationship between the sum of the j=0,1 characters at k=2 and the two ones at k=1 is unravelled.Comment: 42 pages; v2: minor modifications and few references added; version to appear in Nucl. Phys.