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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2408.07139 |
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| _version_ | 1866914220827738112 |
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| author | Yang, Shangjie |
| author_facet | Yang, Shangjie |
| contents | 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 $h_N(x) := \cos\left( (x-1/2)π/N \right)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_07139 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The spectral gap and principle eigenfunction of the random conductance model in a line segment Yang, Shangjie Probability Mathematical Physics 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 $h_N(x) := \cos\left( (x-1/2)π/N \right)$. |
| title | The spectral gap and principle eigenfunction of the random conductance model in a line segment |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2408.07139 |