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Reliability Polynomials and their Asymptotic Limits for Families of Graphs | | Shu-Chiuan Chang
; Robert Shrock
; | | Date: |
28 Aug 2002 | | Journal: | J. Stat. Phys. 112, 1019-1077 (2003) | | Subject: | Statistical Mechanics; Mathematical Physics | cond-mat.stat-mech math-ph math.MP | | Abstract: | We present exact calculations of reliability polynomials $R(G,p)$ for lattice strips $G$ of fixed widths $L_y le 4$ and arbitrarily great length $L_x$ with various boundary conditions. We introduce the notion of a reliability per vertex, $r({G},p) = lim_{|V| o infty} R(G,p)^{1/|V|}$ where $|V|$ denotes the number of vertices in $G$ and ${G}$ denotes the formal limit $lim_{|V| o infty} G$. We calculate this exactly for various families of graphs. We also study the zeros of $R(G,p)$ in the complex $p$ plane and determine exactly the asymptotic accumulation set of these zeros ${cal B}$, across which $r({G})$ is nonanalytic. | | Source: | arXiv, cond-mat/0208538 | | Services: | Forum | Review | PDF | Favorites | |
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