1,720,983 research outputs found
Random walks in a random (fluctuating) environment.
No full textAbstract. The main purpose of this paper is to prove the central limit
theorem for the position at large times of a particle performing a discretetime
random walk on the lattice Zd when the particle interacts with a random
‘environment’ (and starts out at a fixed point of the lattice). Two
cases are considered for the distribution of the particle position: first, the
distribution when the configuration of the ‘environment’ (that is, of the
random field) is fixed at all points of the ‘space-time’ Zd+1 (the so-called
quenched model), and, second, the distribution induced by the joint evolution
of the environment and the particle position under the assumption
that the pair forms a Markov chain (the annealed model). Two cases are
considered also for the quenched model: the values of the field at all points
of ‘space-time’ are independent and identically distributed, or the values of
the field at different times are linked by a homogeneous Markov chain.
In the case of quenched models the central limit theorem with one and the
same limit law is true for almost all configurations of the ‘environment’,
and in the case of annealed models it is true for any initial distribution of
the field. Besides the central limit theorem, the paper briefly treats some
other topics related to these models (decay of correlations, large deviations,
‘the field from the viewpoint of a particle’, and so on)
Directed polymers up to the L_2 threshold
No full textWe present a new technique for studying a general model of discrete-time directed polymers on the lattice Z^d, d≥ 3, in an i.i.d. random medium in the range of the parameters where the L_2-norm of the partition function converges. The method is based on the study of the analytic properties of a kind of complex generating function. We obtain a simple general proof of the classical results, such as convergence of the partition function and diffusion (Theorems 2.1 and 2.2), which were so far obtained for particular cases, and derive some new ones on convergence of higher moments (Theorem 2.3) and on the rate of divergence of the L_2-norm of the partition function at the critical point
Interacting random walk of two particles in a dynamical random environment. Decay of correlations.
No full textThis paper continues the study of a family of models studied earlier by the authors. Two particles perform discrete-time symmetric random walks on the d-dimensional
integer lattice Z^d and interact locally with each other and with a random field (the “environment”) which is indexed by the lattice points. The environment evolves randomly in time its law of evolution is locally affected by the particles .
The whole system is Markovian, and all interactions are assumed to be sufficiently small.
It is shown that the correlations of the field at two fixed points decay in time as C t^{(−d/2)−1}. Under additional assumptions the constant C may be expressed to first
order as the sum of the corresponding constants for the one-particle model. The proofs are based on the analysis of the spectrum of the system’s transition operator. The
results may be extended to models containing a finite number of particles
Directed polymers in Markov random media
No full textWe consider a model of directed polymers in discrete space and time assuming a Markov dependence of the environment in time. We extend results on the almost-sure validity of
the central limit theorem for small randomness in space dimension 3 which were previously obtained for an independent environment by relying on two main technical tools: the analysis of the spectrum of a kind of transfer matrix which allows one to treat the averaged model, and the
explicit construction of a multiplicative orthonormal basis in the appropriate L_2 space, together with cluster estimates of cumulants of the basis functions
Central limit theorem for a random walk in dynamical environment: integral and local.
No full textIn the paper a model of a random walk on the d-dimensional lattice Z^d, d = 1, 2, . . . , in a dynamic environment
which is i.i.d. in space-time is considered. The environment is described by a field which locally takes values in a finite set.
We prove that:
- If the stochastic term is small the central limit theorem holds almost surely, with the same
parameters as for the random walk with averaged transition probabilities (averaged RW).
- The leading term in the asymptotics for large t differs from the corresponding term for the
averaged walk by a factor depending on the field “as seen from the final point”
Interacting random walk in a dynamical random environment. II. The environment "from the point of view of the particle".
No full textFor the same model as in the paper I we now consider the "environment from the point of view of the random walk", which is a field with Markov evolution. We prove that as time grows its distribution tends to a limit which is absolutely continuous with respect to the unperturbed equilibrium distributions. Its correlations decay for d≥ 3 as e^{-\al t}\over t^{d/2}
Central Limit Theorem for the Random Walk of One and Two Particles in a Random Environment with Mutual Interaction
No full text
Almost-sure Central Limit Theorem for a Model of Random Walk in Fluctuating Random Environment
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