Geometry of the Ising persistence problem and the universal Bonnet-Manin Painlev\'e VI distribution

TL;DR

This paper derives the full persistence probability distribution for a non-Markovian process related to spin systems, revealing a connection to Painlevé VI equations and geometric surfaces, and providing exact formulas and interpretations.

Contribution

It establishes a link between the persistence distribution, Painlevé VI equations, and Bonnet surfaces, offering a novel geometric perspective on the Ising persistence problem.

Findings

Persistence distribution governed by Painlevé VI system.

Exact Pfaffian decomposition into Fredholm determinants.

Persistence exponent as asymptotic mean curvature of Bonnet surfaces.

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 Painlev\'e VI system arising from an exact Fredholm Pfaffian structure associated with the integrable sech kernel, $K_{\mathrm{sech}}=1/(2 \pi \cosh[(x-y)/2])$. 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 Painlev\'e VI equation. The corresponding Painlev\'e VI connection problem determines the persistence exponent as a limiting value at infinity. We further show that the Painlev\'e 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 Painlev\'e 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.