313 research outputs found
Properties of the limit shape for some last passage growth models in random environments
We study directed last passage percolation on the first quadrant of the
planar square lattice whose weights have general distributions, or
equivalently, ./G/1 queues in series. The service time distributions of the
servers vary randomly which constitutes a random environment for the model.
Equivalently, each row of the last passage model has its own randomly chosen
weight distribution. We investigate the limiting time constant close to the
boundary of the quadrant. Close to the y-axis, where the number of random
distributions averaged over stays large, the limiting time constant takes the
same universal form as in the homogeneous model. But close to the x-axis we see
the effect of the tail of the distribution of the random means attached to the
rows.Comment: 24 pages, this paper has been accepted for publication in Stochastic
Processes and their Application
Current fluctuations of a system of one-dimensional random walks in random environment
We study the current of particles that move independently in a common static
random environment on the one-dimensional integer lattice. A two-level
fluctuation picture appears. On the central limit scale the quenched mean of
the current process converges to a Brownian motion. On a smaller scale the
current process centered at its quenched mean converges to a mixture of
Gaussian processes. These Gaussian processes are similar to those arising from
classical random walks, but the environment makes itself felt through an
additional Brownian random shift in the spatial argument of the limiting
current process.Comment: Published in at http://dx.doi.org/10.1214/10-AOP537 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Large deviation rate functions for the partition function in a log-gamma distributed random potential
We study right tail large deviations of the logarithm of the partition
function for directed lattice paths in i.i.d. random potentials. The main
purpose is the derivation of explicit formulas for the -dimensional
exactly solvable case with log-gamma distributed random weights. Along the way
we establish some regularity results for this rate function for general
distributions in arbitrary dimensions.Comment: Published in at http://dx.doi.org/10.1214/12-AOP768 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Quenched invariance principle for multidimensional ballistic random walk in a random environment with a forbidden direction
We consider a ballistic random walk in an i.i.d. random environment that does
not allow retreating in a certain fixed direction. We prove an invariance
principle (functional central limit theorem) under almost every fixed
environment. The assumptions are nonnestling, at least two spatial dimensions,
and a moment for the step of the walk uniformly in the
environment. The main point behind the invariance principle is that the
quenched mean of the walk behaves subdiffusively.Comment: Published at http://dx.doi.org/10.1214/009117906000000610 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Geometric RSK correspondence, Whittaker functions and symmetrized random polymers
We show that the geometric lifting of the RSK correspondence introduced by
A.N. Kirillov (2001) is volume preserving with respect to a natural product
measure on its domain, and that the integrand in Givental's integral formula
for GL(n,R)-Whittaker functions arises naturally in this context. Apart from
providing further evidence that Whittaker functions are the natural analogue of
Schur polynomials in this setting, our results also provide a new
`combinatorial' framework for the study of random polymers. When the input
matrix consists of random inverse gamma distributed weights, the probability
distribution of a polymer partition function constructed from these weights can
be written down explicitly in terms of Whittaker functions. Next we restrict
the geometric RSK mapping to symmetric matrices and show that the volume
preserving property continues to hold. We determine the probability law of the
polymer partition function with inverse gamma weights that are constrained to
be symmetric about the main diagonal, with an additional factor on the main
diagonal. The third combinatorial mapping studied is a variant of the geometric
RSK mapping for triangular arrays, which is again showed to be volume
preserving. This leads to a formula for the probability distribution of a
polymer model whose paths are constrained to stay below the diagonal. We also
show that the analogues of the Cauchy-Littlewood identity in the setting of
this paper are equivalent to a collection of Whittaker integral identities
conjectured by Bump (1989) and Bump and Friedberg (1990) and proved by Stade
(2001, 2002). Our approach leads to new `combinatorial' proofs and
generalizations of these identities, with some restrictions on the parameters.Comment: v2: significantly extended versio
The strict-weak lattice polymer
We introduce the strict-weak polymer model, and show the KPZ universality of
the free energy fluctuation of this model for a certain range of parameters.
Our proof relies on the observation that the discrete time geometric q-TASEP
model, studied earlier by A. Borodin and I. Corwin, scales to this polymer
model in the limit q->1. This allows us to exploit the exact results for
geometric q-TASEP to derive a Fredholm determinant formula for the strict-weak
polymer, and in turn perform rigorous asymptotic analysis to show KPZ scaling
and GUE Tracy-Widom limit for the free energy fluctuations. We also derive
moments formulae for the polymer partition function directly by Bethe ansatz,
and identify the limit of the free energy using a stationary version of the
polymer model.Comment: 23 pages, 4 figure
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