Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Geometry of the Ising persistence problem and the universal Bonnet-Manin Painlevé VI distribution

Ivan Dornic, Robert Conte

Abstract

We determine the full persistence probability distribution for a non-Markovian stochastic process, motivated by first-passage questions arising in interacting spin systems and allied systems. We show that this distribution is governed by a distinguished Painlevé VI system arising from an exact Fredholm Pfaffian structure associated with the integrable sech kernel, $K_{\mathrm{sech}}=1/(2 π\cosh[(x-y)/2])$. The universal persistence exponent originally obtained by Derrida, Hakim and Pasquier is recovered as an asymptotic observable and acquires a natural geometric interpretation. In the stationary scaling regime, the persistence probability admits an exact Pfaffian decomposition into even and odd Fredholm determinants of the integrable \emph{sech} kernel. These determinants are controlled by a unique global solution of a second-order nonlinear ordinary differential equation, which is identified as a particular Painlevé VI equation. The corresponding Painlevé VI connection problem determines the persistence exponent as a limiting value at infinity. We further show that the Painlevé VI system governing persistence admits a direct geometric interpretation: the relevant solution coincides with the mean curvature of a one-parameter family of Bonnet surfaces immersed in $\mathbb R^3$. A folding transformation between such surfaces singles out the Painlevé VI equation with Manin coefficients $[0,0,0,0]$, which in particular governs the universal persistence distribution in the symmetric Ising case. In this framework, the persistence exponent is identified with the asymptotic mean curvature of the associated surface.

Geometry of the Ising persistence problem and the universal Bonnet-Manin Painlevé VI distribution

Abstract

We determine the full persistence probability distribution for a non-Markovian stochastic process, motivated by first-passage questions arising in interacting spin systems and allied systems. We show that this distribution is governed by a distinguished Painlevé VI system arising from an exact Fredholm Pfaffian structure associated with the integrable sech kernel, . The universal persistence exponent originally obtained by Derrida, Hakim and Pasquier is recovered as an asymptotic observable and acquires a natural geometric interpretation. In the stationary scaling regime, the persistence probability admits an exact Pfaffian decomposition into even and odd Fredholm determinants of the integrable \emph{sech} kernel. These determinants are controlled by a unique global solution of a second-order nonlinear ordinary differential equation, which is identified as a particular Painlevé VI equation. The corresponding Painlevé VI connection problem determines the persistence exponent as a limiting value at infinity. We further show that the Painlevé VI system governing persistence admits a direct geometric interpretation: the relevant solution coincides with the mean curvature of a one-parameter family of Bonnet surfaces immersed in . A folding transformation between such surfaces singles out the Painlevé VI equation with Manin coefficients , which in particular governs the universal persistence distribution in the symmetric Ising case. In this framework, the persistence exponent is identified with the asymptotic mean curvature of the associated surface.

Paper Structure

This paper contains 15 sections, 6 theorems, 166 equations, 1 table.

Table of Contents

  1. Introduction
  2. Statement of results
  3. Relation to existing literature
  4. Outline of the paper
  5. Integrable structure of persistence probability: proof strategy
  6. Fredholm or Pfaffians determinants and gap probabilities
  7. Integrable structure and closed ODE system obeyed by the resolvent kernels
  8. Integration in terms of a system of intertwined $\mathrm{P}_{\mathrm{VI}}$ functions
  9. The distinguished Painlevé VI governing persistence
  10. Transcendental nature of the persistence distribution
  11. Conclusion
  12. The DHP formula for the persistence probability as a Pfaffian gap-spacing probability with the sech kernel
  13. The Painlevé VI solutions for the mean curvature and the metric of Bonnet surfaces in ${\mathbb{R}}^3$
  14. Three remarkable second order nonlinear ODEs: $\mathrm{P}_{\mathrm{VI}}, H_{\mathrm{VI}}, \mathrm{C}_{\mathrm{VI}}$
  15. The Borodin-Okounkov formula and the solution of the Bonnet $\mathrm{P}_{\mathrm{VI}}$ connection problem

Key Result

Theorem 1.1

In the stationary scaling regime and starting from $m$-magnetized random initial conditions, the persistence probability $P_0^+(\ell;m)$ that the Ising spin located at the origin of semi-infinite chain evolving with zero-temperature Glauber dynamics has always been in the $+$ state for a length of t Here, ${\mathcal{D}}^\pm= \mathcal{D}^\pm(\ell;\xi)$, with $\xi=\xi(m)=1-m^2$, are the Fredholm det

Theorems & Definitions (10)

  • Theorem 1.1: Pfaffian decomposition and Fredholm representation
  • Theorem 1.2: Painlevé VI structure of persistence and global solution of Bonnet surfaces
  • proof : First proof
  • proof : Second proof
  • Proposition B.1
  • Proposition B.2
  • proof
  • proof
  • Proposition C.1
  • Proposition D.1