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Counting self-avoiding walks in a bounded region
... To determine these values and bounds, we use transfer matrix methods. These have been used extensively in statistical mechanics [1, 6] and combinatorics (see, e.g., [5, 7]) to evaluate and analyse many different functions defined on lattice graphs of various kinds. In statistical mechanics, they have been used to study partition functions [1, 6]. ...
... In statistical mechanics, they have been used to study partition functions [1, 6]. In graph theory, they have been used to count structures such as colourings [5] and self-avoiding walks [7]. There has been a number of attempts to estimate the number of legal Go positions on square lattice boards including the standard 19 × 19 one. ...
We use transfer matrix methods to determine bounds for the numbers of legal Go positions for various numbers of players on some planar lattice graphs, including square lattice graphs such as those on which the game is normally played. We also Þnd bounds on limiting constants that describe the behaviour of the number of legal positions on these lattice graphs as the dimensions of the lattices tend to inÞnity. These results amount to giving bounds for some speciÞc evaluations of Go polynomials on these graphs.
... That paper with Donnelly considered antivoter models, in which the vertices of a graph are each given a colour Black or White, with the colour on a vertex randomly chosen to be biased away from whichever colour is more frequent on neighbouring vertices. Some of Keith Edwards's work also harked back to Dominic's early interest in self-avoiding walks on square-lattice graphs [27]. ...
We review the work of Dominic Welsh (1938-2023), tracing his remarkable influence through his theorems, expository writing, students, and interactions. He was particularly adept at bringing different fields together and fostering the development of mathematics and mathematicians. His contributions ranged widely across discrete mathematics over four main career phases: discrete probability, matroids and graphs, computational complexity, and Tutte-Whitney polynomials. We give particular emphasis to his work in matroid theory and Tutte-Whitney polynomials.
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