Mirror Model and Critical Percolation

Abstract

The mirror model on the two-dimensional square lattice is a random walk that begins at the origin. For the first step, the walker randomly chooses one of four directions, north, east, south or west. After the first step, the walker only turns either left or right with equal probability but cannot go straight or backwards at each site. If the walker returns to a site that has been visited before, it has to make the same turn as the previous visiting time. The walk stops if it revisits the first bond. Note that the walk is not allowed to revisit the bond that has been visited before, and no site is visited more than twice. We refer to this walk model as the mirror model. Since for each vertex in Z2, we can position a two-sided mirror, either a north-west mirror or a north-east mirror, with probability 1/2 to correspond the turn that the walk makes at the site. For any walk starting at the origin, we can find the corresponding mirror configuration associated with it. In the first part of the dissertation, we describe its relation to interfaces for bond percolation on the square lattice. We then study the nature of the mirror model process on the square lattice, and use some percolation results to show localization, self-touching property, and some other properties that the mirror model has. This random walk model is related to bond percolation, and critical percolation cluster interfaces are conjectured to converge to Schramm-Loewner evolution with $\kappa$ = 6 (SLE6). So we believe that the scaling limit of the mirror model on the square lattice as the lattice spacing going to zero is SLE6. We test this conjecture with numerical simulations of the model and find a good agreement with predictions of SLE6. This is the second part of the dissertation. The last part is devoted to another random walk model – loop erased mirror model (LEMM). We consider mirror model on the square lattice in a bounded simply connected domain that starts and ends at two prescribed boundary points. The walk can revisit a site it has been before and hence form a loop. The LEMM that we study is the simple path defined by erasing these loops in chronological order. Many two- dimensional critical lattice model are believed to have conformally invariant scaling limit. The new loop-erased process is arising from the mirror model, and mirror model is related to critical bond percolation, so we conjecture that the scaling limit of LEMM is conformally invariant. We test this conjecture with Monte Carlo simulations of LEMM and find strong support for the conjecture.