A Note on (Non-)Monotonicity in Temperature for the Ising Model

#### O. Haggstrom

1996, v.2, Issue 4, 529-538

ABSTRACT

We consider the Ising model on a finite graph $G$ at reciprocal temperature $\beta$, and in particular we consider the event $C_{x\leftrightarrow y}$ that two vertices $x$ and $y$ are in the same spin-cluster, and the random variable $C_x$ which is the size of the spin-cluster containing $x$. By means of a simple counterexample, we show that the probability of $C_{x\leftrightarrow y}$ need not be increasing in $\beta$, and that $C_x$ need not be stochastically increasing in $\beta$. On the other hand, a sufficient condition on $G$ for these monotonicities to hold is established.

Keywords: Ising model,stochastic monotonicity,random-cluster representation,percolation