Date of Award
Level of Access Assigned by Author
Doctor of Philosophy (PhD)
Peter H. Kleban
Second Committee Member
William O. Bray
Third Committee Member
R. Dean Astumian
This thesis explores critical two-dimensional percolation in bounded regions in the continuum limit. The main method which we employ is conformal field theory (CFT). Our specific results follow from the null-vector structure of the c = 0 CFT that applies to critical two-dimensional percolation. We also make use of the duality symmetry obeyed at the percolation point, and the fact that percolation may be understood as the g-state Potts model in the limit q —»• 1. Our first results describe the correlations between points in the bulk and boundary intervals or points, i.e. the probability that the various points or intervals are in the same percolation cluster. These quantities correspond to order-parameter profiles under the given conditions, or cluster connection probabilities. We consider two specific cases: an anchoring interval, and two anchoring points. We derive results for these and related geometries using the CFT null-vectors for the corresponding boundary condition changing (bcc) operators. In addition, we exhibit several exact relationships between these probabilities. These relations between the various bulk-boundary connection probabilities involve parameters of the CFT called operator product expansion (OPE) coefficients. We then compute several of these OPE coefficients, including those arising in our new probability relations. Beginning with the familiar CFT operator 4>i,2, which corresponds to a free-fixed spin boundary change in the g-state Potts model, we then develop physical interpretations of the bcc operators. We argue that, when properly normalized, higher-order bcc operators correspond to successive fusions of multiple (f)^ operators. Finally, by identifying the derivative of 01 2 with the operator it±, we derive several new quantities called first crossing densities. These new results are then combined and integrated to obtain the three previously known crossing quantities in a rectangle: the probability of a horizontal crossing cluster, the probability of a cluster crossing both horizontally and vertically, and the expected number of horizontal crossing clusters. These three results were known to be solutions to a certain fifth-order differential equation, but until now no physically meaningful explanation had appeared. This differential equation arises naturally in our derivation.
Harris Simmons, Jacob Joseph, "Applications of Conformal Field Theory to Problems in 2D Percolation" (2007). Electronic Theses and Dissertations. 1104.
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