Hamiltonian cycles on bicolored random planar maps

Abstract

We study the statistics of Hamiltonian cycles on various families of bicolored random planar maps (with the spherical topology). These families fall into two groups corresponding to two distinct universality classes with respective central charges $c=-1$ and $c=-2$. The first group includes generic $p$-regular maps with vertices of fixed valency $p\ge 3$, whereas the second group comprises maps with vertices of mixed valencies, and the so-called rigid case of $2q$-regular maps ($q\ge 2$) for which, at each vertex, the unvisited edges are equally distributed on both sides of the cycle. We predict for each class its universal configuration exponent $\gamma$, as well as a new universal critical exponent $\nu$ characterizing the number of long-distance contacts along the Hamiltonian cycle. These exponents are theoretically obtained by using the Knizhnik, Polyakov and Zamolodchikov (KPZ) relations, with the appropriate values of the central charge, applied, in the case of $\nu$, to the corresponding critical exponent on regular (hexagonal or square) lattices. These predictions are numerically confirmed by analyzing exact enumeration results for $p$-regular maps with $p=3,4,\dots ,7$, and for maps with mixed valencies $(2,3)$, $(2,4)$ and $(3,4)$.

arXiv comment41 pages, 19 figures, 9 tables

References in corpus (22)

• The dimension of the SLE curves
• A Guide to Stochastic Loewner Evolution and its Applications
• Conformally Invariant Fractals and Potential Theory
• Path Crossing Exponents and the External Perimeter in 2D Percolation
• Two-Dimensional Critical Percolation: The Full Scaling Limit
• Duality of Chordal SLE
• Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model
• Operator Spectrum and Exact Exponents of the Fully Packed Loop Model
• Field theory of compact polymers on the square lattice
• Duality and KPZ in Liouville Quantum Gravity
• Exact Results for Hamiltonian Walks from the Solution of the Fully Packed Loop Model on the Honeycomb Lattice
• Two perspectives of the 2D unit area quantum sphere and their equivalence
• Mating of trees for random planar maps and Liouville quantum gravity: a survey
• A recursive approach to the O(n) model on random maps via nested loops
• Schramm Loewner Evolution and Liouville Quantum Gravity
• Geometrically constrained statistical systems on regular and random lattices: From folding to meanders
• The packing of two species of polygons on the square lattice
• Hamiltonian Cycles on Random Eulerian Triangulations
• Higher Conformal Multifractality
• Permutons, meanders, and SLE-decorated Liouville quantum gravity
• Exponents for Hamiltonian paths on random bicubic maps and KPZ
• Components in meandric systems and the infinite noodle