The spectral gap of two dimensional Ising model with a hole: Shrinking effect of contours

Abstract

We consider the spectral gap of the two demensional Ising model on a special graph with some special boundary conditions. The special graph is a finite square with a square hole on its center part, that is, we consider a finite square of side $2L +1$, and we remove another smaller finite square of side $2L_{1} +1(L_{1} <L)$ which has the same center as the finite square of side $2L +1$, therefore there is a square hole at the center part of the finite square of side $2L +1$, we denote this special graph by $\Lambda (L,L_{1})$. On this graph, the boundary of $\Lambda (L,L_{1})$ is composed of “inner boundary” and “outer boundary”. We will discuss two different boundary conditions of the Ising model on $\Lambda (L,L_{1})$, one is that the outer boundary condition is plus and minus spins’ “mixed” boundary condition and the inner boundary condition is plus boundary condition; the other is that the outer boundary condition is plus boundary condition and the inner boundary condition is an arbitrary boundary condition. On above two different boundary conditions, in the absence of an external field and at large inverse temperature $\beta$ we will show the upper bound of the spectral gap of Ising model for the first of above boundary conditions, and the lower bound of the spectral gap of Ising model for the second of above boundary conditions. These two results show that if we consider the first of above boundary conditions, and exchange this inner boundary condition with the outer boundary condition of the Ising model on $\Lambda (L,L_{1})$, the spectral gap of Ising model will be greately changed. The results can be extended to some other cases, for example, we can consider some other boundary conditions and some other graphs.