2017/12/30 by Béatrice de Tilière, de Tilière, Béatrice · 1 citation
Physics and Astronomy · Mathematics · #Theoretical and Computational Physics #Markov Chains and Monte Carlo Methods #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.1801.00207
Consider an elliptic parameter k; we introduce a family of Zu-Dirac\noperators (\K(u))_u\∈ Re( mathbbT(k)), relate them to the\nZ-massive Laplacian of [BdTR17b], and extend to the full Z-invariant case\nthe results of Kenyon [Ken02] on discrete holomorphic and harmonic functions,\nwhich correspond to the case k=0. We prove, in a direct statistical mechanics\nway, how and why the Zu-Dirac and Z-massive Laplacian operators appear in\nthe Z-invariant Ising model, considering the case of infinite and finite\nisoradial graphs. More precisely, consider the dimer model on the Fisher graph\n\G^ scriptscriptstyle\F arising from a Z-invariant\nIsing model. We express coefficients of the inverse Fisher Kasteleyn operator\nas a function of the inverse Zu-Dirac operator and also as a function of the\nZ-massive Green function; in particular this proves a (massive) random walk\nrepresentation of important observables of the Ising model. We prove that the\nsquared partition function of the Ising model is equal, up to a constant, to\nthe determinant of the Z-massive Laplacian operator with specific boundary\nconditions, the latter being the partition function of rooted spanning forests.\nTo show these results, we relate the inverse Fisher Kasteleyn operator and that\nof the dimer model on the bipartite graph\n\G^ scriptscriptstyle\Q arising from the XOR-Ising\nmodel, and we prove matrix identities between the Kasteleyn matrix of\n\G^ scriptscriptstyle\Q and the Zu-Dirac operator,\nthat allow to reach inverse matrices as well as determinants.\n