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Limit Theorems for Random Walk Local Time, Bootstrap Percolation and Permutation Statistics


Limit theorems are established in three different contexts. The first one concerns exceptional points of the simple random walk in planar lattice domains approximating a given bounded continuum domain D of R^2 with wired boundary conditions; the walk is run for a time proportional to the expected cover time. The sets of suitably defined thick, thin, light and avoided points are shown to be asymptotically distributed according to a log-normal multiple of the zero-average Liouville Quantum Gravity measure in D.

The second area of interest concerns the scaling limit of 2-neighbor polluted bootstrap percolation on Z^2. Here each site is initially independently declared polluted with probability q, occupied with probability p, and vacant otherwise. At each step, each vacant site becomes occupied by contact with 2 or more occupied neighbors. It is shown that in the limit when p, q ↓ 0 with q/p^2 → λ [0,∞) , the regime of small λ results in asymptotic full occupancy while the regime of large \lambda results in asymptotic full vacancy of the terminal configuration. The proof is based on an identification of a continuum percolation model of blocking contours where these regimes correspond to absence and presence of ordinary percolation, respectively.

The last area of interest concerns the number of descents and peaks in a given conjugacy class of a random permutation of n elements. Asymptotic normality is proved proved in the limit n→∞ for suitably scaled versions of these quantities by establishing a uniform estimate on their moment generating functions.

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