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

TL;DR

This work analyzes the spectral gap and the principal eigenfunction of a one-dimensional random walk in a random conductance model on the segment $[1,N]$ with Neumann boundaries. Under a Law of Large Numbers type condition on the resistances, the gap scales as $\mathrm{gap}_N=(1+o(1))\pi^2/N^2$ and the principal eigenfunction $g_N$ with $g_N(1)=1$ is uniformly close to the cosine profile $h_N(x)=\cos((x-1/2)\pi/N)$, with precise control on the weighted discrete derivative. The authors also establish sharp monotonicity properties of the eigenfunctions: $g_1$ is strictly monotone and for $j\ge2$, $g_j$ is $j$-monotone. Intuitively, a tangent-ODE scaling argument for a rescaled recursion yields the critical $\alpha=\pi^2$ that governs the asymptotics, and a perturbative, homogenization-based analysis provides explicit convergence rates and shape results. Collectively, these results give explicit 1D disordered spectral data and enable sharp mixing-time bounds for related particle systems, illustrating how disorder yields the same leading-order spectral behavior as the homogeneous case while providing detailed eigenfunction profiles.

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 $\mathrm{gap}_{N}=(1+o(1))π^2/N^2$ and the principle eigenfunction $g_N$ with $g_N(1)=1$ corresponding to the spectral gap is well approximated by $h_N(x) := \cos\left( (x-1/2)π/N \right)$.

Figures (2)

• Figure 1: A graphical explanation for the random walk in conductances $(c(x, x+1))_{1 \le x<N}$: at edge $\{x, x+1 \}$ there is a Poisson clock with rate $c(x, x+1)>0$ for all $1\le x<N$, and we swap the contents of the two sites $x, \, x+1$ when the clock on the edge $\{x, x+1\}$ rings.
• Figure 2: In the figures above, solid lines depict the function $\Xi^{(c)}(b,\lambda)$ with $\lambda>0$ fixed, the black dashed lines stand for $y=b$, and the blue dashes lines are $b=c$. There are three cases about the number of fixed points of the map $b \mapsto \Xi^{(c)}(\lambda, b)$ according to the relation between $\lambda$ and $4c$ as shown above.

Theorems & Definitions (35)

• Proposition 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Remark 1.5
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3