Vertex-reinforced random walks and a conjecture of Pemantle (original) (raw)

Vertex reinforced random walks with exponential interaction on complete graphs

2020

We describe a model for m vertex reinforced interacting random walks on complete graphs with d ≥ 2 vertices. The transition probability of a random walk to a given vertex depends exponentially on the proportion of visits made by all walks to that vertex. The individual proportion of visits is modulated by a strength parameter that can be set equal to any real number. This model covers a large variety of interactions including different vertex repulsion and attraction strengths between any two random walks as well as self-reinforced interactions. We show that the process of empirical vertex occupation measures defined by the interacting random walks converges (a.s.) to the limit set of the flow induced by a smooth vector field. Further, if the set of equilibria of the field is formed by isolated points, then the vertex occupation measures converge (a.s.) to an equilibrium of the field. These facts are shown by means of the construction of a strict Lyapunov function. We show that if t...

On properties of non-markovian random walk in one dimension

Journal of Physics: Conference Series, 2021

We study a strongly Non-Markovian variant of random walk in which the probability of visiting a given site i is a function f of number of previous visits v(i) to the site. If the probability is inversely proportional to number of visits to the site, say f(i)=1/(v(i)+1)α the probability distribution of visited sites tends to be flat for α>0 compared to simple random walk. For f(i)=exp(-v(i)), we observe a distribution with two peaks. The origin is no longer the most probable site. The probability is maximum at site k(t) which increases in time. For f(i)=exp(-v(i)) and for α>0 the properties do not change as the walk ages. However, for α<0, the properties are similar to simple random walk asymptotically. We study lattice covering time for these functions. The lattice covering time scales as Nz, with z=2, for α ≤ 0, z>2 for a >0 and z<2 for f(i)=exp(-v(i)).

Dynamics of vertex-reinforced random walks

The Annals of Probability, 2011

We generalize a result from Volkov (2001,[23]) and prove that, on a large class of locally finite connected graphs of bounded degree (G, ∼) and symmetric reinforcement matrices a = (a i,j ) i,j∈V (G) , the vertex-reinforced random walk (VRRW) eventually localizes with positive probability on subsets which consist of a complete d-partite subgraph with possible loops plus its outer boundary.

On self-attracting random walks

Proceedings of Symposia in Pure Mathematics, 1994

In this survey paper we mainly discuss the results contained in two of our recent articles [2] and [5]. Let {X t } t≥0 be a continuous-time, symmetric, nearest-neighbour random walk on Z d. For every T > 0 we define the transformed path measure d P T = (1/Z T) exp(H T) dP, where P is the original one and Z T is the appropriate normalizing constant. The Hamiltonian H T imparts the self-attracting interaction of the paths up to time T. We consider the case where H T is given by a potential function V on Z d with finite support, and the case H T = −N T , where N T denotes the number of points visited by the random walk up to time T. In both cases the typical paths under P T as T → ∞ clump together much more than those of the free random walk and give rise to localization phenomena. 1991 Mathematics Subject Classification. 60F10 (primary), 60F05 and 60K35 (secondary). Key words and phrases. large deviations-attractive interaction-d-dimensional random walk-maximum entropy principle-weak convergence-localization-confinement. Research supported by the Swiss National Foundation, Contract Nr. 21-298333.90. This is a survey paper. Details of the proofs of the results are published in [2] and [5].

Random walks on the finite components of

2005

The expected n-step return-probability EµP o [ ˆ Xn = o] of a random walk ˆ Xn with symmetric transition probabilities on a random partial graph of a regular graph G of degree δ with transitive automorphism group Aut(G) is considered. The law µ of the random edge-set is assumed to be stationary with respect to some transitive, unimodular subgroup Γ of Aut(G). By the spectral theory of finite random walks, using interlacing techniques, bounds in terms of functionals of the cluster size are obtained:

Two Results on Asymptotic Behaviour of Random Walks in Random Environment

2016

Two results on Asymptotic Behaviour of Random Walks in Random Environment Jeremy Voltz Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2016 In the first chapter of this thesis, we consider a model of directed polymer in 1 + 1 dimensions in a product-type random environment ω(t, x) = b(t)F (x), where the fields F and b are i.i.d., with F (x) continuous, symmetric and bounded, and b(t) = ±1 with probabilty 1/2. Thus ω can be viewed as the field F oscillating in time. We consider directed last-passage percolation through this random field; namely, we investigate the behavior of the length n polymer path with maximal action, where the action of a path is simply the sum of the environment variables it moves through. We prove a law of large numbers for the maximal action of the path from the origin to a fixed endpoint (n, bαnc), and investigate the limiting shape function a(α). We prove that this shape function is non-linear, and has a corner at α = 0, thus i...

Occupation time limit theorems for independent random walks

1994

We summarize many limit theorems for systems of independent simple random walks in Z. These theorems are classi ed into four classes: weak convergence, moderate deviations, large deviations and enormous deviations. A hierarchy of relations is pointed out and some open problems are posed. Extensions to function spaces are also mentioned.

On fixation of activated random walks

Arxiv preprint arXiv:0910.3730, 2009

We prove that for the Activated Random Walks model on transitive unimodular graphs, if there is fixation, then every particle eventually fixates, almost surely. We deduce that the critical density is at most 1.

On a directionally reinforced random walk

Proceedings of the American Mathematical Society, 2014

We consider a generalized version of a directionally reinforced random walk, which was originally introduced by Mauldin, Monticino, and von Weizsäcker. Our main result is a stable limit theorem for the position of the random walk in higher dimensions. This extends a result of Horváth and Shao that was previously obtained in dimension one only (however, in a more stringent functional form).