Exact Persistence Exponent for the2D-Diffusion Equation and Related Kac Polynomials

Abstract

We compute the persistence for the 2d-diffusion equation with random initial condition, i.e., the probability p0(t) that the diffusion field, at a given point x in the plane, has not changed sign up to time t. For large t, we show that p0(t) ∼ t −θ(2) with θ(2) = 3/16. Using the connection between the 2d-diffusion equation and Kac random polynomials, we show that the probability q0(n) that Kac polynomials, of (even) degree n, have no real root decays, for large n, as q0(n) ∼ n −3/4 . We obtain this result by using yet another connection with the truncated orthogonal ensemble of random matrices. This allows us to compute various properties of the zero-crossings of the diffusing field, equivalently of the real roots of Kac polynomials. Finally, we unveil a precise connection with a fourth model: the semi-infinite Ising spin chain with Glauber dynamics at zero temperature. arXiv:1806.11275v1 [cond-mat.stat-mech]

Persistence Exponent for the Simple Diffusion Equation: The Exact Solution for Any Integer Dimension

“…In [3], the authors have defined a correlation function ( ) C T , just like [1] [2], to carry the calculation forward. Let us also consider [6] where the authors have used Kac Polynomials [19] to obtain the "exact exponent" in 2d. The answer obtained agrees perfectly with [1] the calculated exponent will be different from the actual value.…”

An infinite-dimensional helix invariant under spherical projections

“…For the process g(t) itself, a similar property was used by Matsumoto and Shirai [7,Lemma 4.2] to establish the Pfaffian character of both real and complex zeroes of g(t). Recently, Poplavskyi and Schehr [9] used the Pfaffian character of the zeroes of X to compute the persistence exponent of X and several related processes.…”

“…Open questions. The property of Gaussian processes studied here was used in [8,7,9] to establish the determinantal/Pfaffian character of the zeroes of the corresponding process. It is natural to ask for a description of all (sufficiently smooth) stationary, centered, Gaussian processes whose zeroes form a Pfaffian/determinantal point process.…”

A generalisation of the relation between zeros of the complex Kac polynomial and eigenvalues of truncated unitary matrices

“…And moreover this determinantal point process is identical to that formed by the eigenvalues of any (N − 1) × (N − 1) sub-block of a random complex unitary matrix chosen with Haar measure in the limit N → ∞ [12]. For a recent application of this latter coincidence to the problem of persistence exponents, see [15]. A different, and more general, coincidence between the distribution of the zeros of a random power series with coefficients independently distributed as standard real Gaussians, and that of the eigenvalues of a particular random matrix ensemble, has been given by Tao [19].…”

“…15) where z 1 = z and z 2 = w. Use of the(see e.g. [4, eq, (4.34)]) known Cauchy double alternant − |w| 2 )(1 − |z| 2 )(1 − zw)(1 − wz) |z| 2 |w| 2 |w − z| 2…”