The spectral gap and principle eigenfunction of the random conductance model in a line segment

Abstract

In this paper, we study the spectral gap and principle eigenfunction of the random walk in the line segment $[1,N]$ with conductances $c^{(N)}(x,x+1{)}_{1\le x<N}$ where $c^{(N)}(x,x+1)>0$ is the rate of the random walk jumping from site $x$ to site $x+1$ and vice versa. Writing $r^{(N)}(x,x+1):=1/c^{(N)}(x,x+1)$, under the assumption \begin{equation*} \limsup_{N\to \infty}\, \frac{1}{N}\sup_{1< m \le N}\, \left| \sum_{x=2}^m r^{(N)}(x-1, x)- (m-1) \right|\;=\;0\,, \end{equation*} we prove that the spectral gap, denoted by ${\mathrm{gap}}_N$, of the process satisfies ${\mathrm{gap}}_N=(1+o(1)){\pi }^2/N^2$ and the principle eigenfunction $g_N$ with $g_N(1)=1$ corresponding to the spectral gap is well approximated by $h_N(x):=\cos \left((x-1/2)\pi /N\right)$.

25 pages, Comments are welcome