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, $$K_{\textrm{sech}}=1/(2 \pi \cosh [(x-y)/2])$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>K</mml:mi> <mml:mtext>sech</mml:mtext> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mi>π</mml:mi> <mml:mo>cosh</mml:mo> <mml:mrow> <mml:mo>[</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>-</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . 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 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 $$\mathbb R^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>R</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:math> . 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.