Asymptotic Spectral Analysis on the Johnson Graphs in Infinite Degree and Zero Temperature Limit

概要

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We discuss a scaling limit of the spectral distribution of the adjacency operator (or Laplacian) on the Johnson graph J(v,d) with respect to the Gibbs state associated with the graph. (The adjacency operator on J(v,d), whose vertices consist of the d-subsets of a v-set, gives us the Bernoulli–Laplace diffusion model.) We compute the limit distribution and its moments exactly in the situation of infinite degree (v,d→∞) and zero temperature (β→∞) limit where the three parameters v, d and β keep appropriate scaling balances. A method of quantum decomposition of an adjacency operator plays a key role for expressing the limit moments in terms of a creation operator, an annihilation operator and a number operator on a suitable Hilbert space. Using this expression, we analyze the limit moments in detail in combinatorial and analytical ways. In our previous work [Hora, A., Probab. Theory Relat. Fields, 118: 115–130 (2000)], a partial solution was given where v=2d and some additional constraints of scaling were assumed. In this note, we remove all such restrictions.