Foremost, I would like to express my sincere gratitude to my supervisor David Cimasoni for the co... more Foremost, I would like to express my sincere gratitude to my supervisor David Cimasoni for the continuous support of my PhD research, for his patience, motivation and enthusiasm. I wish I could do my PhD one more time under his supervision. Besides my advisor, I would like to thank the rest of my thesis committee: Professor Yvan Velenik and Professor Béatrice de Tilière, for accepting to be in the jury of my thesis. My thanks also goes to Section of Mathematics of University of Geneva which gave me a very good opportunity to work and finish my thesis. I also thank my fellows in the section for useful discussions, for the time we were working together and for all the fun we have had in the last four years. Last but not least, I would like to thank my wife, my little daughter and my family members for supporting me throughout my PhD study.
We consider the monomer-dimer model on weighted graphs embedded in surfaces with boundary, with t... more We consider the monomer-dimer model on weighted graphs embedded in surfaces with boundary, with the restriction that only monomers located on the boundary are allowed. We give a Pfaffian formula for the corresponding partition function, which generalises the one obtained by Giuliani, Jauslin and Lieb for graphs embedded in the disk [9]. Our proof is based on an extension of a bijective method mentioned in [9], together with the Pfaffian formula for the dimer partition function of Cimasoni-Reshetikhin [3].
Journal of Statistical Mechanics: Theory and Experiment, 2020
We give a Pfaffian formula to compute the partition function of the Ising model on any graph G em... more We give a Pfaffian formula to compute the partition function of the Ising model on any graph G embedded in a closed, possibly non-orientable surface. This formula, which is suitable for computational purposes, is based on the relation between the Ising model on G and the dimer model on its terminal graph G T. By combining the ideas of Loebl-Masbaum [14], Tesler [18], Cimasoni [2, 3] and Chelkak-Cimasoni-Kassel [1], we give an elementary proof for the formula.
Journal of Statistical Mechanics: Theory and Experiment, 2016
Given a weighted graph G embedded in a non-orientable surface Σ, one can consider the correspondi... more Given a weighted graph G embedded in a non-orientable surface Σ, one can consider the corresponding weighted graph G embedded in the so-called orientation cover Σ of Σ. We prove identities relating twisted partition functions of the dimer model on these two graphs. When Σ is the Möbius strip or the Klein bottle, then Σ is the cylinder or the torus, respectively, and under some natural assumptions, these identities imply relations between the genuine dimer partition functions Z(G) and Z(G). For example, we show that if G is a locally but not globally bipartite graph embedded in the Möbius strip, then Z(G) is equal to the square of Z(G). This extends results for the square lattice previously obtained by various authors.
Given a weighted graph G embedded in a non-orientable surface Σ, one can consider the correspondi... more Given a weighted graph G embedded in a non-orientable surface Σ, one can consider the corresponding weighted graph G ∼ embedded in the so-called orientation cover Σ of Σ. We prove identities relating twisted partition functions of the dimer model on these two graphs. When Σ is the Möbius strip or the Klein bottle, then Σ is the cylinder or the torus, respectively, and under some natural assumptions, these identities imply relations between the genuine dimer partition functions Z(G ) and Z G ( ) ∼
Foremost, I would like to express my sincere gratitude to my supervisor David Cimasoni for the co... more Foremost, I would like to express my sincere gratitude to my supervisor David Cimasoni for the continuous support of my PhD research, for his patience, motivation and enthusiasm. I wish I could do my PhD one more time under his supervision. Besides my advisor, I would like to thank the rest of my thesis committee: Professor Yvan Velenik and Professor Béatrice de Tilière, for accepting to be in the jury of my thesis. My thanks also goes to Section of Mathematics of University of Geneva which gave me a very good opportunity to work and finish my thesis. I also thank my fellows in the section for useful discussions, for the time we were working together and for all the fun we have had in the last four years. Last but not least, I would like to thank my wife, my little daughter and my family members for supporting me throughout my PhD study.
We consider the monomer-dimer model on weighted graphs embedded in surfaces with boundary, with t... more We consider the monomer-dimer model on weighted graphs embedded in surfaces with boundary, with the restriction that only monomers located on the boundary are allowed. We give a Pfaffian formula for the corresponding partition function, which generalises the one obtained by Giuliani, Jauslin and Lieb for graphs embedded in the disk [9]. Our proof is based on an extension of a bijective method mentioned in [9], together with the Pfaffian formula for the dimer partition function of Cimasoni-Reshetikhin [3].
Journal of Statistical Mechanics: Theory and Experiment, 2020
We give a Pfaffian formula to compute the partition function of the Ising model on any graph G em... more We give a Pfaffian formula to compute the partition function of the Ising model on any graph G embedded in a closed, possibly non-orientable surface. This formula, which is suitable for computational purposes, is based on the relation between the Ising model on G and the dimer model on its terminal graph G T. By combining the ideas of Loebl-Masbaum [14], Tesler [18], Cimasoni [2, 3] and Chelkak-Cimasoni-Kassel [1], we give an elementary proof for the formula.
Journal of Statistical Mechanics: Theory and Experiment, 2016
Given a weighted graph G embedded in a non-orientable surface Σ, one can consider the correspondi... more Given a weighted graph G embedded in a non-orientable surface Σ, one can consider the corresponding weighted graph G embedded in the so-called orientation cover Σ of Σ. We prove identities relating twisted partition functions of the dimer model on these two graphs. When Σ is the Möbius strip or the Klein bottle, then Σ is the cylinder or the torus, respectively, and under some natural assumptions, these identities imply relations between the genuine dimer partition functions Z(G) and Z(G). For example, we show that if G is a locally but not globally bipartite graph embedded in the Möbius strip, then Z(G) is equal to the square of Z(G). This extends results for the square lattice previously obtained by various authors.
Given a weighted graph G embedded in a non-orientable surface Σ, one can consider the correspondi... more Given a weighted graph G embedded in a non-orientable surface Σ, one can consider the corresponding weighted graph G ∼ embedded in the so-called orientation cover Σ of Σ. We prove identities relating twisted partition functions of the dimer model on these two graphs. When Σ is the Möbius strip or the Klein bottle, then Σ is the cylinder or the torus, respectively, and under some natural assumptions, these identities imply relations between the genuine dimer partition functions Z(G ) and Z G ( ) ∼
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