Universal Painlevé 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 Painlevé VI transcendent, the persistence exponent θ(m)/2 emerging as an asymptotic decay rate. Using a known yet remarkable correspondence that relates Painlevé 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 in terms of Gauss intrinsic curvature. Since the same sechkernel with an underlying Pfaffian structure shows up in a variety of Gaussian first-passage problems, our Painlevé VI provides their universal first-passage probability distribution, in a manner exactly analogous to the famous Painlevé II Tracy-Widom laws. The tail behavior in the magnetizationsymmetric case of our full scaling function allows to recover the exact persistence exponent θ(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. 121, 150601 (2018)]. Our Painlevé VI tau-function persistence distribution also bears a correspondence with a c = 1 conformal field theory, the monodromy parameters giving the dimensions of the associated primary fields. This yields θ(0) = 3β, with β = 1/8 the Onsager-Yang magnetization exponent for the critical 2d Ising model, a plausible conjecture relating a nonequilibrium exponent to ordinary static critical behavior in one more space dimension, that suggests more generally that methods of boundary conformal field theory should be helpful in determining the critical properties of other unsolved nonequilibrium 1d processes.

Record statistics of integrated random walks and the random acceleration process

“…The limit of these exponents, lim k→∞ θ k = 3/16 = 0.1875, is related to the diffusion equation in the plane. Its exact value has been derived only recently [42,43]. Here, too, the number of records is expected to grow linearly for all k ≥ 2, with a universal, kdependent, asymptotic probability of record breaking growing from r ∞ = 0.387 627 .…”

Asymptotic expansions for a class of Fredholm Pfaffians and interacting particle systems

“…The factor φ p = (−t −1 log(4p(1 − p))) 1/2 and (32) follows from (13). For κ 2 (p), we use the expressions in Theorem 1, where all the integrals can be evaluated using the explicit Gaussian densities ρ * n .…”

“…A common feature of the Pfaffian point processes related to Kac polynomials, truncated orthogonal random matrices and exit measures for interacting particle systems, is the appearing of the scalar sech-kernel; see Section 2 for a review of "derived" Pfaffian point processes and the corresponding scalar kernels. Exploiting the integrability of the sech-kernel, Ivan Dornic analyses the asymptotics of a distinguished solution to Painlevé VI equation to rederive the formula for the empty interval probability for the real roots of Kac polynomials, [13]. Interestingly enough, the integrable structure of kernels associated with the single time law of CABM and the real Ginibre ensemble is not at all obvious.…”

Sharp asymptotics for Fredholm Pfaffians related to interacting particle systems and random matrices

“…A common feature of the Pfaffian point processes related to Kac polynomials, truncated orthogonal random matrices and exit measures for interacting particle systems is the appearing of the scalar sech-kernel, see Section 2 for a review of 'derived' Pfaffian point processes and the corresponding scalar kernels. Exploiting the integrability of the sech-kernel, Ivan Dornic analyses the asymptotics of solutions to Painlevé VI equation to re-derive the formula for the empty interval probability for the real roots of Kac polynomials, [12]. Interestingly enough, the integrable structure of kernels associated with the single time law of CABM and the real Ginibre ensemble is not at all obvious.…”