8 research outputs found
Lyapunov exponents of Green's functions for random potentials tending to zero
We consider quenched and annealed Lyapunov exponents for the Green's function
of , where the potentials , are i.i.d.
nonnegative random variables and is a scalar. We present a
probabilistic proof that both Lyapunov exponents scale like as
tends to 0. Here the constant is the same for the quenched as for
the annealed exponent and is computed explicitly. This improves results
obtained previously by Wei-Min Wang. We also consider other ways to send the
potential to zero than multiplying it by a small number.Comment: 16 pages, 3 figures. 1 figure added, very minor corrections. To
appear in Probability Theory and Related Fields. The final publication is
available at http://www.springerlink.com, see
http://www.springerlink.com/content/p0873kv68315847x/?p=4106c52fc57743eba322052bb931e8ac&pi=21
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
Positive temperature versions of two theorems on first-passage percolation
The estimates on the fluctuations of first-passsage percolation due to
Talagrand (a tail bound) and Benjamini--Kalai--Schramm (a sublinear variance
bound) are transcribed into the positive-temperature setting of random
Schroedinger operators.Comment: 15 pp; to appear in GAFA Seminar Note
Variational formulas and cocycle solutions for directed polymer and percolation models
We discuss variational formulas for the law of large numbers limits of certain models of motion in a random medium: namely, the limiting time constant for last-passage percolation and the limiting free energy for directed polymers. The results are valid for models in arbitrary dimension, steps of the admissible paths can be general, the environment process is ergodic under spatial translations, and the potential accumulated along a path can depend on the environment and the next step of the path. The variational formulas come in two types: one minimizes over gradient-like cocycles, and another one maximizes over invariant measures on the space of environments and paths. Minimizing cocycles can be obtained from Busemann functions when these can be proved to exist. The results are illustrated through 1+1 dimensional exactly solvable examples, periodic examples, and polymers in weak disorder
On Range and Local Time of Many-dimensional Submartingales
We consider a discrete-time process adapted to some filtration which lives on
a (typically countable) subset of , . For this process,
we assume that it has uniformly bounded jumps, is uniformly elliptic (can
advance by at least some fixed amount with respect to any direction, with
uniformly positive probability). Also, we assume that the projection of this
process on some fixed vector is a submartingale, and that a stronger additional
condition on the direction of the drift holds (this condition does not exclude
that the drift could be equal to 0 or be arbitrarily small). The main result is
that with very high probability the number of visits to any fixed site by time
is less than for some . This in its turn implies
that the number of different sites visited by the process by time should be
at least .Comment: 23 pages, 8 figures; to appear in Journal of Theoretical Probabilit