Random Planar Lattices Research Articles

Isoperimetric inequalities in the Brownian plane

We consider the model of the Brownian plane, which is a pointed noncompact random metric space with the topology of the complex plane. The Brownian plane can be obtained as the scaling limit in distribution of the uniform infinite planar triangulation or the uniform infinite planar quadrangulation and is conjectured to be the universal scaling limit of many others random planar lattices. We establish sharp bounds on the probability of having a short cycle separating the ball of radius r centered at the distinguished point from infinity. Then we prove a strong version of the spatial Markov property of the Brownian plane. Combining our study of short cycles with this strong spatial Markov property we obtain sharp isoperimetric bounds for the Brownian plane.

Counting Coloured Planar Maps: Differential Equations

We address the enumeration of q-coloured planar maps counted by the number of edges and the number of monochromatic edges. We prove that the associated generating function is differentially algebraic, that is, satisfies a non-trivial polynomial differential equation with respect to the edge variable. We give explicitly a differential system that characterizes this series. We then prove a similar result for planar triangulations, thus generalizing a result of Tutte dealing with their proper q-colourings. In statistical physics terms, we solve the q-state Potts model on random planar lattices. This work follows a first paper by the same authors, where the generating function was proved to be algebraic for certain values of q, including q=1, 2 and 3. It is known to be transcendental in general. In contrast, our differential system holds for an indeterminate q. For certain special cases of combinatorial interest (four colours; proper q-colourings; maps equipped with a spanning forest), we derive from this system, in the case of triangulations, an explicit differential equation of order 2 defining the generating function. For general planar maps, we also obtain a differential equation of order 3 for the four-colour case and for the self-dual Potts model.

Sums of random matrices and the Potts model on random planar maps

We compute the partition function of the q-states Potts model on a random planar lattice with allowed, equally weighted colours on a connected boundary. To this end, we employ its matrix model representation in the planar limit, generalising a result by Voiculescu for the addition of random matrices to a situation beyond free probability theory. We show that the partition functions with p and q − p colours on the boundary are related algebraically. Finally, we investigate the phase diagram of the model when and comment on the conformal field theory description of the critical points.

Monomer-dimer problem on random planar honeycomb lattice

We consider the monomer-dimer (MD) problem on a random planar honeycomb lattice model, namely, the random multiple chain. This is a lattice system with non-periodic boundary condition, whose generating process is inspired by the growth of single walled zigzag carbon nanotubes. By applying algebraic and combinatorial techniques we establish a calculating expression of the MD partition function for bipartite graphs, which corresponds to the permanent of a matrix. Further, by using the transfer matrix argument we show that the computing problem of the permanent of high order matrix can be converted into some lower order matrices for this family of lattices, based on which we derive an explicit recurrence formula for evaluating the MD partition function of multiple chains and random multiple chains. Finally, we analyze the expectation of the number of monomer-dimer arrangements on a random multiple chain and the asymptotic behavior of the annealed MD entropy when the multiple chain becomes infinite in width and length, respectively.

Duality analysis on random planar lattices

The conventional duality analysis is employed to identify a location of a critical point on a uniform lattice without any disorder in its structure. In the present study, we deal with the random planar lattice, which consists of the randomized structure based on the square lattice. We introduce the uniformly random modification by the bond dilution and contraction on a part of the unit square. The random planar lattice includes the triangular and hexagonal lattices in extreme cases of a parameter to control the structure. A modern duality analysis fashion with real-space renormalization is found to be available for estimating the location of the critical points with a wide range of the randomness parameter. As a simple test bed, we demonstrate that our method indeed gives several critical points for the cases of the Ising and Potts models and the bond-percolation thresholds on the random planar lattice. Our method leads to not only such an extension of the duality analyses on the classical statistical mechanics but also a fascinating result associated with optimal error thresholds for a class of quantum error correction code, the surface code on the random planar lattice, which is known as a skillful technique to protect the quantum state.

Counting colored planar maps: Algebraicity results

We address the enumeration of properly q-colored planar maps, or more precisely, the enumeration of rooted planar maps M weighted by their chromatic polynomial χ M ( q ) and counted by the number of vertices and faces. We prove that the associated generating function is algebraic when q ≠ 0 , 4 is of the form 2 + 2 cos ( j π / m ) , for integers j and m. This includes the two integer values q = 2 and q = 3 . We extend this to planar maps weighted by their Potts polynomial P M ( q , ν ) , which counts all q-colorings (proper or not) by the number of monochromatic edges. We then prove similar results for planar triangulations, thus generalizing some results of Tutte which dealt with their proper q-colorings. In statistical physics terms, the problem we study consists in solving the Potts model on random planar lattices. From a technical viewpoint, this means solving non-linear equations with two “catalytic” variables. To our knowledge, this is the first time such equations are being solved since Tutteʼs remarkable solution of properly q-colored triangulations.

Spanning Forests on Random Planar Lattices

The generating function for spanning forests on a lattice is related to the q-state Potts model in a certain q→0 limit, and extends the analogous notion for spanning trees, or dense self-avoiding branched polymers. Recent works have found a combinatorial perturbative equivalence also with the (quadratic action) O(n) model in the limit n→−1, the expansion parameter t counting the number of components of the forest.

On the Asymptotic Number of Plane Curves and Alternating Knots

We present a conjecture for the power-law exponent in the asymptotic number of types of plane curves as the number of self-intersections goes to infinity. In view of the description of prime alternating links as flype equivalence classes of plane curves, a similar conjecture is made for the asymptotic number of prime alternating knots. The rationale leading to these conjectures is given by quantum field theory. Plane curves are viewed as configurations of loops on random planar lattices, that are in turn interpreted as a model of two-dimensional quantum gravity with matter. The identification of the universality class of this model yields the conjecture. Since approximate counting or sampling planar curves with more than a few dozens of intersections is an open problem, direct confrontation with numerical data yields no convincing indication on the correctness of our conjectures. However, our physical approach yields a more general conjecture about connected systems of curves. We take advantage of this to design an original and feasible numerical test, based on recent perfect samplers for large planar maps. The numerical data strongly support our identification with a conformal field theory recently described by Readand Saleur.

Random planar lattices and integrated superBrownian excursion

In this paper, a surprising connection is described between a specific brand of random lattices, namely planar quadrangulations, and Aldous’ Integrated SuperBrownian Excursion (ISE). As a consequence, the radius r n of a random quadrangulation with n faces is shown to converge, up to scaling, to the width r=R−L of the support of the one-dimensional ISE, or precisely: More generally the distribution of distances to a random vertex in a random quadrangulation is described in its scaled limit by the random measure ISE shifted to set the minimum of its support in zero. The first combinatorial ingredient is an encoding of quadrangulations by trees embedded in the positive half-line, reminiscent of Cori and Vauquelin’s well labelled trees. The second step relates these trees to embedded (discrete) trees in the sense of Aldous, via the conjugation of tree principle, an analogue for trees of Vervaat’s construction of the Brownian excursion from the bridge. From probability theory, we need a new result of independent interest: the weak convergence of the encoding of a random embedded plane tree by two contour walks to the Brownian snake description of ISE. Our results suggest the existence of a Continuum Random Map describing in term of ISE the scaled limit of the dynamical triangulations considered in two-dimensional pure quantum gravity.

Critical probability and scaling functions of bond percolation on two-dimensional random lattices

We locate the critical probability of bond percolation on two-dimensional random lattices as . Because of the symmetry with respect to permutation of the two axes for random lattices, we expect that for an aspect ratio of unity and sufficiently large lattices, the probability of horizontal spanning equals the probability of vertical spanning. This is confirmed by our Monte Carlo simulations. We show that the ideas of universal scaling functions and non-universal metric factors can be extended to random lattices by studying the existence probability and the percolation probability P on finite square, planar triangular, and random lattices with periodic boundary conditions using a histogram Monte Carlo method. Our results also indicate that the metric factors may be the same between random lattices and planar triangular lattices provided that the aspect ratios are 1 and .

The two-dimensional O(2) model on a random planar lattice at strong coupling

The large spacing phase of the infinite random matrix chain, which represents the strongly coupled two-dimensional O(2) model on a random planar lattice, is explored. A class of solutions valid for large lattice spacings is constructed. It is proved that these solutions exhibit the critical exponents characteristic of pure two-dimensional gravity. The character expansion for the chain model is developed and an order parameter governing the Kosterliz-Thouless phase transition is identified.

Exactly solvable Potts models, bond- and tree-like percolation on dynamical (random) planar lattice

It is shown that q-state Potts models on dynamical planar (random) lattice (DPL) are exactly solvable: the problem is reduced to some large N saddle point equations. Further simplications occur in the cases of the Ising model on DPL (q = 2, solved in [1], of bond percolation on random quenced Lattice (q → 1) and of tree-like percolation on DPL (q → 0). For the last one the exact free energy is calculated and the existence of phase transition is shown.

The ising model on a random planar lattice: The structure of the phase transition and the exact critical exponents

We investigate the critical properties of a recently proposed exactly soluble Ising model on a planar random dynamical lattice representing a regularization of the zero-dimensional string with internal fermions. The sum over all lattices gives rise to a new quantum degree of freedom - fluctuation of the metric. The whole system of critical exponents is found: α = −1, β = 1 2 , γ = 2, δ = 5, ν· D = 3. To test the universality we have used the planar graphs with the coordination number equal to 4 (θ 4 theory graphs) as well as with the equal to 3 (θ 3 theory graphs or triangulations). The critical exponents coincide for both cases.