Shape transition under excess self-intersections for transient random walk. (English) Zbl 1202.60151

The paper extends the previous line of research of the present author on random walks in a random scenery. Special attention is paid to large deviations for two-fold self-intersection local times of a transient random walk in \(d\)-dimensions, with \(d\geq 3\). For a simple random walk on \(\mathbb{Z}^d\) started at the origin, the basic technical input is the \(q\)-norm i.e. the sum of \(q\)-th power of the local times which, for integer \(q\), can be written in terms of the \(q\)-fold self-intersection local times of that walk. The shape transition is described as the parameter \(q\) crosses the critical value \(q_c= d/(d-2)\). That is quantified in terms of two different sites exploration strategy by a transient random walk. A central limit theorem is established for the \(q\)-norm of the local times in \(d\geq 4\).


60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J25 Continuous-time Markov processes on general state spaces
82C22 Interacting particle systems in time-dependent statistical mechanics
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