Universal Painlev\'e VI Probability Distribution in Pfaffian Persistence and Gaussian First-Passage Problems with a sech-Kernel

Abstract

We recast the persistence probability for the spin located at the origin of a half-space arbitrarily $m$-magnetized Glauber-Ising chain as a Fredholm Pfaffian gap probability generating function with a sech-kernel. This is then spelled out as a tau-function for a certain Painlev\'e VI transcendent, the persistence exponent $\theta(m)/2$ emerging as an asymptotic decay rate. Using a known yet remarkable correspondence that relates Painlev\'e equations to Bonnet surfaces, the persistence probability also acquires a geometric meaning in terms of the mean curvature of the latter, and even a topological one at the magnetization-symmetric point. Since the same sech-kernel with an underlying Pfaffian structure shows up in a variety of Gaussian first-passage problems, our Painlev\'e VI provides their universal first-passage probability distribution, in a manner exactly analogous to the famous Painlev\'e II Tracy-Widom laws. The tail behavior in the magnetization-symmetric case of our full scaling function allows to recover the exact persistence exponent $\theta(0)/2=3/16$ for the $2d$-diffusing random field or for random real Kac's polynomials, a particular result found very recently by Poplavskyi and Schehr (Phys. Rev. Lett. {\bf 121}, 150601 (2018)). Our Painlev\'e VI tau-function characterization of the persistence probability also bears a correspondence with a $c=1$ conformal field theory, the monodromy parameters giving the dimensions of the associated primary fields. Thereby $\theta(0)/2=3 \beta/2$, with $\beta=1/8$ the Onsager-Yang magnetization exponent for the critical $2d$ Ising model. This relates a nonequilibrium exponent to ordinary static critical behavior in one more space dimension, and suggests more generally that methods of boundary conformal field theory should be helpful for determining the critical properties of other unsolved nonequilibrium $1d$ processes.