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Self-avoiding walk on nonunimodular transitive graphs. (English) Zbl 1448.60187
Summary: We study self-avoiding walk on graphs whose automorphism group has a transitive nonunimodular subgroup. We prove that self-avoiding walk is ballistic, that the bubble diagram converges at criticality, and that the critical two-point function decays exponentially in the distance from the origin. This implies that the critical exponent governing the susceptibility takes its mean-field value, and hence that the number of self-avoiding walks of length \(n\) is comparable to the \(n\) th power of the connective constant. We also prove that the same results hold for a large class of repulsive walk models with a self-intersection based interaction, including the weakly self-avoiding walk. All of these results apply in particular to the product \(T_k\times\mathbb{Z}^d\) of a \(k\)-regular tree \((k\geq 3)\) with \(\mathbb{Z}^d \), for which these results were previously only known for large \(k\).

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
60G50 Sums of independent random variables; random walks
05C30 Enumeration in graph theory
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
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