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 $(3,4)$.

41 pages, 19 figures, 9 tables

References in corpus (9)

• A Guide to Stochastic Loewner Evolution and its Applications
• 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
• Duality and KPZ in Liouville Quantum Gravity
• Geometrically constrained statistical systems on regular and random lattices: From folding to meanders
• 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