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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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