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- W2810104937 abstract "In the loop $O(n)$ model a collection of mutually-disjoint self-avoiding loops is drawn at random on a finite domain of a lattice with probability proportional to $${lambda^{# mbox{edges}} n^{# mbox{loops}},}$$ where $lambda, n in [0, infty)$. Let $mu$ be the connective constant of the lattice and, for any $n in [0, infty)$, let $lambda_c(n)$ be the largest value of $lambda$ such that the loop length admits uniformly bounded exponential moments. It is not difficult to prove that $lambda_c(n) =1/mu$ when $n=0$ (in this case the model corresponds to the self-avoiding walk) and that for any $n geq 0$, $lambda_c(n) geq 1/mu$. In this note we prove that, begin{align*} lambda_c(n) & > 1/mu , , , , , , , , , , , mbox{whenever $n >0$}, lambda_c(n) & geq 1/mu , + , c_0 , n , + , O(n^2), end{align*} on $mathbb{Z}^d$, with $d geq 2$, and on the hexagonal lattice, where $c_0>0$. This means that, when $n$ is positive (even arbitrarily small), as a consequence of the mutual repulsion between the loops, a phase transition can only occur at a strictly larger critical threshold than in the self-avoiding walk." @default.
- W2810104937 created "2018-07-10" @default.
- W2810104937 creator A5011864453 @default.
- W2810104937 date "2018-06-25" @default.
- W2810104937 modified "2023-09-27" @default.
- W2810104937 title "Shifted critical threshold in the loop $O(n)$ model at arbitrary small $n$" @default.
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