arXiv subjects
Publications and source records attributed to Mikhail Menshikov.
We study a one-dimensional random walk whose expected drift depends both on time and the position of a particle. We establish a non-trivial phase transition for the recurrence vs. transience of the walk, and show some interesting applications to Friedman's urn, as well as showing the connection with Lamperti's walk with asymptotically zero drift.
We study a model of a polling system, that is, a collection of $d$ queues with a single server that switches from queue to queue. The service time distribution and arrival rates change randomly every time a queue is emptied. This model is mapped to a mathematically equivalent model of a random walk with random choice of transition probabilities, a model which is of independent interest. All our results are obtained using methods from the constructive theory of Markov chains. We determine conditions for the existence of polynomial moments of hitting times for the random walk. An unusual phenomenon of thickness of the region of null recurrence for both the random walk and the queueing model is also proved.
We report on the asymptotic behaviour of a new model of random walk, we term the bindweed model, evolving in a random environment on an infinite multiplexed tree. The term \textit{multiplexed} means that the model can be viewed as a nearest neighbours random walk on a tree whose vertices carry an internal degree of freedom from the finite set $\{1,...,d\}$, for some integer $d$. The consequence of the internal degree of freedom is an enhancement of the tree graph structure induced by the replacement of ordinary edges by multi-edges, indexed by the set $\{1,...,d\}\times\{1,...,d\}$. This indexing conveys the information on the internal degree of freedom of the vertices contiguous to each edge. The term \textit{random environment} means that the jumping rates for the random walk are a family of edge-indexed random variables, independent of the natural filtration generated by the random variables entering in the definition of the random walk; their joint distribution depends on the index of each component of the multi-edges. We study the large time asymptotic behaviour of this random walk and classify it with respect to positive recurrence or transience in terms of a specific parameter of the probability distribution of the jump rates. This classifying parameter is shown to coincide with the critical value of a matrix-valued multiplicative cascade on the ordinary tree (\textit{i.e.} the one without internal degrees of freedom attached to the vertices) having the same vertex set as the state space of the random walk. Only results are presented here since the detailed proofs will appear elsewhere.
We study random walks in a random environment on a regular, rooted, coloured tree. The asymptotic behaviour of the walks is classified for ergodicity/transience in terms of the geometric properties of the matrix describing the random environment. A related problem, with only one type of vertices and quite stringent conditions on the transition probabilities but on general trees has been considered previously in the literature LyoPem. In the presentation we give here, we restrict the study of the process on a regular graph instead of the irregular graph used in LyoPem. The close connection between various problems on random walks in random environment and the so called multiplicative chaos martingale is underlined by showing that the classification of the random walk problem can be drawn by the corresponding classification for the multiplicative chaos, at least for those situations where both problems have been solved by independent methods. The chaos counterpart of the problem we considered here has not yet been solved. The results we obtain for the random walk problem localise the position of the critical point. We conjecture that the additional conditions needed for the chaos problem to have non trivial solutions will be the same as the ones needed for the random walk to be null recurrent.
If one randomly punches holes in an infinite tensed membrane, when does the tension cease to exist? This problem was introduced by R. Connelly in connection with applications of rigidity theory to natural sciences. We outline a mathematical theory of tension based on graph rigidity theory and introduce several probabilistic models for this problem. We show that if the ``centers'' of the holes are distributed in R^2 according to Poisson law with parameter λ>0, and the distribution of sizes of the holes is independent of the distribution of their centers, the tension vanishes on all of R^2 for any value of λ. In fact, it follows from a more general result on the behavior of iterative convex hulls of connected subsets of R^d, when the initial configuration of subsets is distributed according to Poisson law and the sizes of the elements of the original configuration are independent of this Poisson distribution. For the latter problem we establish the existence of a critical threshold in terms of the number of iterative convex hull operations required for covering all of R^d. The processes described in the paper are somewhat related to bootstrap and rigidity percolation models.