an:01971039
Zbl 1118.82301
Chang, Shu-Chiuan; Shrock, Robert
Reliability polynomials and their asymptotic limits for families of graphs
EN
J. Stat. Phys. 112, No. 5-6, 1019-1077 (2003).
00097956
2003
j
82B20 05B35 05C35
Reliability polynomial; Potts model; Tutte polynomial
Summary: We present exact calculations of reliability polynomials \(R(G,p)\) for lattice strips \(G\) of fixed widths \(L_y \leq 4\) and arbitrarily great length \(L_x\) with various boundary conditions. We introduce the notion of a reliability per vertex,
\[
r(\{G\},p)=\lim_ {| V| \to\infty}R(G,p)^ {1/| V| },
\]
where \(| V| \) denotes the number of vertices in \(G\) and \(\{G\}\) denotes the formal limit \(\lim_ {| V| \to\infty}G\). 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 \(\mathcal B\), across which \(r(\{G\})\) is nonanalytic.