The spectrum of the random environment and localization of noise

We consider random walk on a mildly random environment on finite transitive d-regular graphs of increasing girth. After scaling and centering, the analytic spectrum of the transition matrix converges in distribution to a Gaussian noise. An interesting phenomenon occurs at d = 2: as the limit graph changes from a regular tree to the integers, the noise becomes localized. The graphs of the noise covariance structure for d = 4, 3, 2.1 from above.     

A simple network of nodes moving on the circle

Two simple Markov processes are examined, one in discrete and one in continuous time, arising from idealized versions of a transmission protocol for mobile networks. We consider two independent walkers moving with constant speed on the discrete or continuous circle, and changing directions at independent geometric (respectively, exponential) times. One of the walkers carries a message that wishes to travel as far and as fast as possible in the clockwise direction. The message stays with its current carrier unless the two walkers meet, the carrier is moving counter‐clockwise, and the other walker is moving clockwise. Then the message jumps to the other walker. Explicit expressions are derived for the long‐term average clockwise speed and number of jumps made of the message, via the solution of associated boundary value problems. The tradeoff between speed and cost (measured as the rate of jumps) is also examined.     

Functional Limit Theorems for the Pólya Urn

Journal of Theoretical Probability - For the plain Pólya urn with two colors, black and white, we prove a functional central limit theorem for the number of white balls, assuming that the...     

Localization of favorite points for diffusion in a random environment

For a diffusion _Xt$X_t$ in a one-dimensional Wiener medium W$W$, it is known that there is a certain process _(br(W))r≥0${\left(b_r\left(W\right)\right)}_{r\ge 0}$ that depends only on the environment, so that _Xt−_blogt(W)$X_t-b_{logt}\left(W\right)$ converges in distribution as t→∞$t\to \infty$. The paths of b$b$ are step functions. Denote by _FX(t)$F_X\left(t\right)$ the point with the most local time for the diffusion at time t$t$. We prove that, modulo a relatively small time change, the paths of the processes _(br(W))r≥0${\left(b_r\left(W\right)\right)}_{r\ge 0}$, _{(FX(er))r≥0}${\left(F_X\left(e^r\right)\right)}_{r\ge 0}$ are close after some large r$r$.