Ballistic Phase of Self-Interacting Random Walks
نویسندگان
چکیده
We explain a unified approach to a study of ballistic phase for a large family of self-interacting random walks with a drift and self-interacting polymers with an external stretching force. The approach is based on a recent version of the OrnsteinZernike theory developed in Campanino et al. (2003, 2004, 2007). It leads to local limit results for various observables (e.g., displacement of the end-point or number of hits of a fixed finite pattern) on paths of n-step walks (polymers) on all possible deviation scales from CLT to LD. The class of models, which display ballistic phase in the “universality class” discussed in the paper, includes self-avoiding walks, DombJoyce model, random walks in an annealed random potential, reinforced polymers and weakly reinforced random walks.
منابع مشابه
Diffusive-Ballistic Transition in Random Walks with Long-Range Self-Repulsion
We prove that a class of random walks on Z2 with long-range selfrepulsive interactions have a diffusive-ballistic phase transition. MSC Numbers: 82B20 82B41 82B26
متن کاملA Note on the Ballistic Limit of Random Motion in a Random Potential
It has been shown that certain types of random walks in random potentials and Brownian motion in Poissonian potentials undergo a phase transition from sub-ballistic to ballistic behavior when the strength of the underlying drift is increased. The ballistic behavior has been manifested by indicating a limiting area for the normalized motion. In the present article, we provide a refined descripti...
متن کاملA combinatorial result with applications to self-interacting random walks
We give a series of combinatorial results that can be obtained from any two collections (both indexed by Z×N) of left and right pointing arrows that satisfy some natural relationship. When applied to certain self-interacting random walk couplings, these allow us to reprove some known transience and recurrence results for some simple models. We also obtain new results for one-dimensional multi-e...
متن کاملDiffusive-ballistic crossover in 1D quantum walks.
We show that particle transport, as characterized by the equilibrium mean square displacement, in a uniform, quantum multibaker map, is generically ballistic in the long time limit, for any fixed value of Planck's constant. However, for fixed times, the semiclassical limit leads to diffusion. Random matrix theory provides explicit analytical predictions for the mean square displacement of a par...
متن کاملAn expansion for self-interacting random walks
We derive a perturbation expansion for general interacting random walks, where steps are made on the basis of the history of the path. Examples of models where this expansion applies are reinforced random walk, excited random walk, the true (weakly) self-avoiding walk and loop-erased random walk. We use the expansion to prove a law of large numbers and central limit theorem for two models: (i) ...
متن کامل