Local Characteristics, Entropy and Limit Theorems for Spanning Trees and Domino Tilings via Transfer-Impedances
| dc.creator | Burton, Robert | |
| dc.creator | Pemantle, Robin | |
| dc.date | 2004-04-02 | |
| dc.date.accessioned | 2026-07-07T05:07:03Z | |
| dc.date.available | 2026-07-07T05:07:03Z | |
| dc.description | Let G be a finite graph or an infinite graph on which Z^d acts with finite fundamental domain. If G is finite, let T be a random spanning tree chosen uniformly from all spanning trees of G; if G is infinite, known methods show that this still makes sense, producing a random essential spanning forest of G. A method for calculating local characteristics (i.e. finite-dimensional marginals) of T from the transfer-impedance matrix is presented. This differs from the classical matrix-tree theorem in that only small pieces of the matrix (n-dimensional minors) are needed to compute small (n-dimensional) marginals. Calculation of the matrix entries relies on the calculation of the Green's function for G, which is not a local calculation. However, it is shown how the calculation of the Green's function may be reduced to a finite computation in the case when G is an infinite graph admitting a Z^d-action with finite quotient. The same computation also gives the entropy of the law of T. These results are applied to the problem of tiling certain lattices by dominos - the so-called dimer problem. Another application of these results is to prove modified versions of conjectures of Aldous on the limiting distribution of degrees of a vertex and on the local structure near a vertex of a uniform random spanning tree in a lattice whose dimension is going to infinity. Included is a generalization of moments to tree-valued random variables and criteria for these generalized moments to determine a distribution. | |
| dc.description | 50 pages | |
| dc.identifier | https://arxiv.org/abs/math/0404048 | |
| dc.identifier | http://arxiv.org/abs/math/0404048 | |
| dc.identifier | Ann. Prob., 21, 1329 - 1371 (1993) | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/70705 | |
| dc.subject | Probability | |
| dc.subject | 60C05, 60K35 | |
| dc.title | Local Characteristics, Entropy and Limit Theorems for Spanning Trees and Domino Tilings via Transfer-Impedances | |
| dc.type | text |