TY - UNPB

T1 - Condensation of random walks and the Wulff crystal

AU - Berestycki, Nathanael

AU - Yadin, Ariel

PY - 2013/5/1

Y1 - 2013/5/1

N2 - 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 $d\ge 3$, 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.

AB - 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 $d\ge 3$, 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.

KW - Mathematics - Probability

KW - Mathematical Physics

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BT - Condensation of random walks and the Wulff crystal

PB - arXiv:1305.0139 [math.PR]

ER -