A Random Walk with Collapsing Bonds and Its Scaling Limit

In this note we introduce a self-interacting random walk in a dynamic random environment, prove that it is recurrent and find its scaling limit. The environment evolves in time in conjunction with the random walk and is nonmarkovian; however we will show that the walk only remembers the recent past. Various models of self-interacting random walks and random walks in dynamic random environments have been studied recently (see for instance [1, 2, 3, 4, 6, 7] and references therein). Consider a particle performing a continuous-time nearest neighbor symmetric random walk on the integers lattice. We assume that the particle is initially at the origin. The times between successive jumps are independent exponential random variables with rate λ. Anytime the particle jumps over the bond connecting two neighboring lattice sites, there is a probability 0 < p ≤ 1 that the bond connecting the sites breaks. The particle is not able to jump over that bond until that bond is repaired. If at the time the particle attempts to jump, one of the bonds neighboring the particle is broken, the particle jumps over the other bond with probability one. If both bonds neighboring the particle are broken, the particle can’t jump when it attempts to do so. The repair times of bonds are independent exponential random variables with rate μ ∈ (0,∞). Initially, there are no broken bonds.