Moderate deviations for the spectral measure of certain random matrices

We derive a moderate deviations principle for matrices of the form XN = DN + WN where WN are Wigner matrices and DN is a sequence of deterministic matrices whose spectral measures converge in a strong sense to a limit µD. Our techniques are based on a dynamical approach introduced by Cabanal-Duvillard and Guionnet

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

What is the correct cost functional for variational data assimilation?

Variational approaches to data assimilation, and weakly constrained four dimensional variation (WC-4DVar) in particular, are important in the geosciences but also in other communities (often under different names). The cost functions and the resulting optimal trajectories may have a probabilistic interpretation, for instance by linking data assimilation with maximum aposteriori (MAP) estimation. This is possible in particular if the unknown trajectory is modelled as the solution of a stochastic differential equation (SDE), as is increasingly the case in weather forecasting and climate modelling. In this situation, the MAP estimator (or “most probable path” of the SDE) is obtained by minimising the Onsager–Machlup functional. Although this fact is well known, there seems to be some confusion in the literature, with the energy (or “least squares”) functional sometimes been claimed to yield the most probable path. The first aim of this paper is to address this confusion and show that the energy functional does not, in general, provide the most probable path. The second aim is to discuss the implications in practice. Although the mentioned results pertain to stochastic models in continuous time, they do have consequences in practice where SDE’s are approximated by discrete time schemes. It turns out that using an approximation to the SDE and calculating its most probable path does not necessarily yield a good approximation to the most probable path of the SDE proper. This suggest that even in discrete time, a version of the Onsager–Machlup functional should be used, rather than the energy functional, at least if the solution is to be interpreted as a MAP estimator

Quenched invariance principle for random walks in balanced random environment

We consider random walks in a balanced random environment in $\mathbb{Z}^d$, $d\geq 2$. We first prove an invariance principle (for $d\ge2$) and the transience of the random walks when $d\ge 3$ (recurrence when $d=2$) in an ergodic environment which is not uniformly elliptic but satisfies certain moment condition. Then, using percolation arguments, we show that under mere ellipticity, the above results hold for random walks in i.i.d. balanced environments.Comment: Published online in Probab. Theory Relat. Fields, 05 Oct 2010. Typo (in journal version) corrected in (26

Anomalous diffusion in disordered multi-channel systems

We study diffusion of a particle in a system composed of K parallel channels, where the transition rates within the channels are quenched random variables whereas the inter-channel transition rate v is homogeneous. A variant of the strong disorder renormalization group method and Monte Carlo simulations are used. Generally, we observe anomalous diffusion, where the average distance travelled by the particle, []_{av}, has a power-law time-dependence []_{av} ~ t^{\mu_K(v)}, with a diffusion exponent 0 \le \mu_K(v) \le 1. In the presence of left-right symmetry of the distribution of random rates, the recurrent point of the multi-channel system is independent of K, and the diffusion exponent is found to increase with K and decrease with v. In the absence of this symmetry, the recurrent point may be shifted with K and the current can be reversed by varying the lane change rate v.Comment: 16 pages, 7 figure

Alternative proof for the localization of Sinai's walk

We give an alternative proof of the localization of Sinai's random walk in random environment under weaker hypothesis than the ones used by Sinai. Moreover we give estimates that are stronger than the one of Sinai on the localization neighborhood and on the probability for the random walk to stay inside this neighborhood

Equality of averaged and quenched large deviations for random walks in random environments in dimensions four and higher

We consider large deviations for nearest-neighbor random walk in a uniformly elliptic i.i.d. environment. It is easy to see that the quenched and the averaged rate functions are not identically equal. When the dimension is at least four and Sznitman's transience condition (T) is satisfied, we prove that these rate functions are finite and equal on a closed set whose interior contains every nonzero velocity at which the rate functions vanish.Comment: 17 pages. Minor revision. In particular, note the change in the title of the paper. To appear in Probability Theory and Related Fields