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Conformally Invariant Observables in the Double-dimer Model

Ghodratipour, Nahid | 2019

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 52742 (04)
  4. University: Sharif University of Technology
  5. Department: Physics
  6. Advisor(s): Rouhani, Shahin
  7. Abstract:
  8. The dimer model studies random dimer covers of a graph. A dimer cover is a subset of the edges of the graph such that each vertex is the endpoint of exactly one edge.The double-dimer model is a natural generalization of the dimer model, whose configuration is a union of two dimer covers, which ends up a set of even-length simple loops and doubled edges.The goal of this thesis is the study of statistical properties of very long doubledimer loops in very large lattices. To this end, we use the Grassmannian representation of the dimer model to compute some loop-related observables in rectangular domains of the square lattice, i.e. 1) the probability distribution of the number of nontrivial loops around the cylinder, which is in agreement with what is obtained for long cylinders by Coulomb gas methods, 2) the expectation value of the number of loops surrounding two points, 3) the left-passage probability, and 4) the connection probabilities of the double-dimer model for four nodes on the boundary. The calculation of the first three observables is based on introducing suitable nontrivial fugacities for certain edges of the lattice, and the fourth one on suitable choices of the boundary monomers in the dimer models constructing the intended double-dimer model. The Grassmannian representation of the dimer model implies the determinantal structure of this model, which makes obtaining the desired coefficients in the expansion of the partition function of the double-dimer model possible.The continuum of the results is consistent with the conjecture of the relation between the double-dimer model and the level-sets of the Gaussian free field, hence with SLE and CLE for = 4. Also, those observables defined purely on the collection of double-dimer loops, offer some tools to study a lattice loop model via conformal loop ensembles. The approach here is close to those common acceptable ones in physics, so makes the rigorous results about the double-dimer model more accessible to physics community
  9. Keywords:
  10. Conformal Invariance ; Schramm-Loewner Evolution (SLE) ; Dimer Models ; Double Dimer Models ; Conformal Invariance ; Conformally Invariant Observables ; Gaussian Free Field ; Conformal Loop Ensemble

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