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We introduce a Gibbs measure on nearest-neighbour paths of length $t$ in the Euclidean $d$-dimensional lattice, where each path is penalised by a factor proportional to the size of its boundary and an inverse temperature $beta$. We prove that, for all $beta>0$, the random walk condensates to a set of diameter $(t/beta)^{1/3}$ in dimension $d=2$, up to a multiplicative constant. In all dimensions $dge 3$, we also prove that the volume is bounded above by $(t/beta)^{d/(d+1)}$ and the diameter is bounded below by $(t/beta)^{1/(d+1)}$. Similar results hold for a random walk conditioned to have local time greater than $beta$ everywhere in its range when $beta$ is larger than some explicit constant, which in dimension two is the logarithm of the connective constant.
We consider branching random walks in $d$-dimensional integer lattice with time-space i.i.d. offspring distributions. This model is known to exhibit a phase transition: If $d ge 3$ and the environment is not too random, then, the total population gro
We consider Activated Random Walks on $Z$ with totally asymmetric jumps and critical particle density, with different time scales for the progressive release of particles and the dissipation dynamics. We show that the cumulative flow of particles thr
We consider symmetric activated random walks on $mathbb{Z}$, and show that the critical density $zeta_c$ satisfies $csqrt{lambda} leq zeta_c(lambda) leq C sqrt{lambda}$ where $lambda$ denotes the sleep rate.
We study a particular class of complex-valued random variables and their associated random walks: the complex obtuse random variables. They are the generalization to the complex case of the real-valued obtuse random variables which were introduced in
We consider branching random walks in $d$-dimensional integer lattice with time-space i.i.d. offspring distributions. When $d ge 3$ and the fluctuation of the environment is well moderated by the random walk, we prove a central limit theorem for the