[Paper Review] Directed animals, quadratic and rewriting systems

Jean‐François Marckert|arXiv (Cornell University)|Dec 5, 2011

Stochastic processes and statistical mechanics15 references3 citations

TL;DR

This paper establishes a deep algebraic connection between the generating function of directed animals on the square lattice—counted by area and perimeter—and a system of quadratic matrix equations. It provides strong evidence that the generating function can be derived from eigenvectors of infinite matrices that arise as fixed points of a rewriting system involving tensor products, offering a novel algebraic framework for solving this long-standing combinatorial problem.

ABSTRACT

A directed animal is a percolation cluster in the directed site percolation model. The aim of this paper is to exhibit a strong relation between the problem of computing the generating function $\G$ of directed animals on the square lattice, counted according to the area and the perimeter, and the problem of solving a system of quadratic equations involving unknown matrices. We present some solid evidence that some infinite explicit matrices, the fixed points of a rewriting like system are the natural solutions to this system of equations: some strong evidence is given that the problem of finding $\G$ reduces to the problem of finding an eigenvector to an explicit infinite matrix. Similar properties are shown for other combinatorial questions concerning directed animals, and for different lattices.

Motivation & Objective

• To establish a formal connection between the generating function of directed animals on the square lattice and a system of quadratic matrix equations.
• To investigate whether infinite matrices, defined as fixed points of a rewriting system using tensor products, provide the natural solution to this matrix system.
• To demonstrate that computing the generating function reduces to finding left and right eigenvectors of these infinite matrices.
• To extend the framework to other lattices and combinatorial models, such as bicolored directed animals and percolation clusters.
• To provide a new algebraic approach to the enumeration of directed animals, potentially enabling the computation of the generating function.

Proposed method

• Formalize the generating function GF_S^G(x,y) for directed animals on a graph G with source S, counting area and perimeter.
• Model the problem using a system of quadratic equations involving unknown finite matrices, derived from recursive decomposition of directed animals.
• Introduce a rewriting system based on tensor products of matrices to generate infinite-size matrices that are candidates for fixed-point solutions.
• Use probabilistic gas models and conditional expectations to derive equations for generating functions, linking them to matrix products and traces.
• Apply Corollary 11 to represent measures on lattice lines using matrix products, enabling the construction of solutions via matrix convergence.
• Extend the method to bicolored models by introducing a three-state gas model with color-dependent transitions, showing its relation to well-colored directed animals.

Experimental results

Research questions

• RQ1 Can the area and perimeter generating function of directed animals on the square lattice be expressed as a solution to a system of quadratic matrix equations?
• RQ2 Do infinite matrices, constructed as fixed points of a rewriting system involving tensor products, serve as natural solutions to this matrix system?
• RQ3 Is the generating function recoverable from the left and right eigenvectors of these infinite matrices?
• RQ4 Can this algebraic framework be generalized to other lattices and combinatorial structures such as bicolored directed animals?
• RQ5 Does the three-state gas model with color-dependent transitions yield a tractable representation for the generating function of well-colored directed animals?

Key findings

• The generating function GF_S^G(x,y) for directed animals on the square lattice is shown to be reducible to the problem of finding eigenvectors of a specific infinite matrix system.
• Strong evidence is provided that the infinite matrices arising as fixed points of the rewriting system are the natural solutions to the quadratic matrix equations governing the generating function.
• The method successfully extends to other lattices and combinatorial models, including the cylindrical lattice and bicolored directed animals.
• For the bicolored model, the generating function GF_{S1,S2}(-p1,-p2) is expressed as a joint probability involving a three-state gas model, linking combinatorics to statistical mechanics.
• The gas model with states {0,1,2} is not Markovian on lines, but its marginals are computable via known Markovian projections, enabling indirect computation of key probabilities.
• The framework provides a universal approach to similar problems, with matrix representations of measures on lattice lines constructed via trace formulas involving products of matrices V^x_(κ) and H^x_(κ).