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Lee-Yang phenomena in edge-coloured graph counting

Maximilian Wiesmann

TL;DR

This work analyzes the zeros of $A^V_n(\lambda)$, a polynomial counting edge-coloured graphs with Euler characteristic $-n$, by expressing it as an exponential integral and performing a Picard–Lefschetz analysis. The main engine is a two-step Lefschetz thimble construction that handles degenerate critical values, yielding a basis for twisted homology and a stationary-phase expansion that explains how zeros accumulate along anti-Stokes curves. Under mild non-degeneracy, the zeros of $A^V_n(\lambda)$ converge to semialgebraic sets defined by equal real parts of metastable energies at critical points, akin to Lee–Yang/Fisher-type phase coexistence. The paper then connects this combinatorial framework to the Ising model on random regular graphs, unifying Lee–Yang and Fisher zeros via two natural specialisations of a two-colour Ising-inspired polynomial.

Abstract

We study the accumulation of zeros of a polynomial arising from the enumeration of edge-coloured graphs along certain limit curves. The polynomial is a variant of an edge-chromatic polynomial, which specialises to the partition function of the ferromagnetic Ising model on a random regular graph. We call this accumulation behaviour a Lee-Yang phenomenon in analogy with the Lee-Yang theorem. The limiting loci are semialgebraic and arise from anti-Stokes curves of an exponential integral.

Lee-Yang phenomena in edge-coloured graph counting

TL;DR

This work analyzes the zeros of , a polynomial counting edge-coloured graphs with Euler characteristic , by expressing it as an exponential integral and performing a Picard–Lefschetz analysis. The main engine is a two-step Lefschetz thimble construction that handles degenerate critical values, yielding a basis for twisted homology and a stationary-phase expansion that explains how zeros accumulate along anti-Stokes curves. Under mild non-degeneracy, the zeros of converge to semialgebraic sets defined by equal real parts of metastable energies at critical points, akin to Lee–Yang/Fisher-type phase coexistence. The paper then connects this combinatorial framework to the Ising model on random regular graphs, unifying Lee–Yang and Fisher zeros via two natural specialisations of a two-colour Ising-inspired polynomial.

Abstract

We study the accumulation of zeros of a polynomial arising from the enumeration of edge-coloured graphs along certain limit curves. The polynomial is a variant of an edge-chromatic polynomial, which specialises to the partition function of the ferromagnetic Ising model on a random regular graph. We call this accumulation behaviour a Lee-Yang phenomenon in analogy with the Lee-Yang theorem. The limiting loci are semialgebraic and arise from anti-Stokes curves of an exponential integral.
Paper Structure (7 sections, 9 theorems, 64 equations, 3 figures)

This paper contains 7 sections, 9 theorems, 64 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Edge-coloured graphs
  3. Lefschetz thimble decomposition
  4. Step 1: construction of Lefschetz thimbles as locally finite chains
  5. Step 2: regularising the Lefschetz thimbles
  6. Accumulation of zeros
  7. Connections to the Ising model

Key Result

Proposition 2.2

The generating function for graphs with marked vertex degrees is Here, $[\boldsymbol{x}^{2\boldsymbol{s}}]$ denotes the coefficient extraction operator, and eq:generating_function is an expression in the formal power series ring $\mathbb{Q}[\Lambda_{\boldsymbol{w}} \,:\, \boldsymbol{w}\in\mathbb{Z}^d_{\geq 0}][[\eta]]$ with coefficients in the formal variables $\

Figures (3)

  • Figure 1: The roots of $A^V_n(\lambda)$ in the complex $\lambda$-plane, for $V(x_1,x_2,\lambda) = \frac{x_1^4}{4!} + \lambda \frac{x_1^2x_2^2}{2!\cdot 2!} + \lambda^2 \frac{x_2^4}{4!}$.
  • Figure 2: Anti-Stokes curves (green and purple) and Stokes curves (grey) for the example depicted in Figure \ref{['fig:roots']}. The roots accumulate along the intersection of the anti-Stokes curves with certain regions in the complement of the Stokes curves (non-shaded regions).
  • Figure 3: An edge-bicoloured graph $G$ with two connected components.

Theorems & Definitions (20)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • Proposition 2.4
  • proof
  • Remark 3.1
  • Theorem 3.2
  • Corollary 3.3
  • proof
  • ...and 10 more