Generalized domino-shuffling
| dc.creator | Propp, James | |
| dc.date | 2001-11-02 | |
| dc.date | 2002-08-29 | |
| dc.date.accessioned | 2026-07-07T04:44:14Z | |
| dc.date.available | 2026-07-07T04:44:14Z | |
| dc.description | The problem of counting tilings of a plane region using specified tiles can often be recast as the problem of counting (perfect) matchings of some subgraph of an Aztec diamond graph A_n, or more generally calculating the sum of the weights of all the matchings, where the weight of a matching is equal to the product of the (pre-assigned) weights of the constituent edges (assumed to be non-negative). This article presents efficient algorithms that work in this context to solve three problems: finding the sum of the weights of the matchings of a weighted Aztec diamond graph A_n; computing the probability that a randomly-chosen matching of A_n will include a particular edge (where the probability of a matching is proportional to its weight); and generating a matching of A_n at random. The first of these algorithms is equivalent to a special case of Mihai Ciucu's cellular complementation algorithm and can be used to solve many of the same problems. The second of the three algorithms is a generalization of not-yet-published work of Alexandru Ionescu, and can be employed to prove an identity governing a three-variable generating function whose coefficients are all the edge-inclusion probabilities; this formula has been used as the basis for asymptotic formulas for these probabilities, but a proof of the generating function identity has not hitherto been published. The third of the three algorithms is a generalization of the domino-shuffling algorithm described by Elkies, Kuperberg, Larsen and Propp; it enables one to generate random ``diabolo-tilings of fortresses'' and thereby to make intriguing inferences about their asymptotic behavior. | |
| dc.description | 18 pages; to appear in Theoretical Computer Science special issue on tilings | |
| dc.identifier | https://arxiv.org/abs/math/0111034 | |
| dc.identifier | http://arxiv.org/abs/math/0111034 | |
| dc.identifier | Theoret. Comput. Sci. 303, no. 2-3, 267--301, Tilings of the plane (2003). | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/62556 | |
| dc.subject | Combinatorics | |
| dc.subject | 05C70; 05C85 | |
| dc.title | Generalized domino-shuffling | |
| dc.type | text |