Exact Persistence Exponent for the 2D-Diffusion Equation and Related Kac Polynomials.

Area of Science:

• Probability theory
• Statistical physics
• Mathematical physics

Background:

• The 2D-diffusion equation models physical phenomena with random initial conditions.
• Understanding the persistence (sign changes) of diffusion fields is crucial in various scientific domains.
• Kac random polynomials and Ising models offer related mathematical frameworks.

Purpose of the Study:

• To compute the probability of sign persistence for a 2D diffusion equation with random initial conditions.
• To establish connections between the diffusion equation, Kac random polynomials, and the Ising spin chain.
• To analyze the behavior of zero crossings in diffusion fields and real roots in Kac polynomials.

Main Methods:

• Calculating the probability p0(t) for large time t using asymptotic analysis.
• Leveraging the established link between the 2D-diffusion equation and Kac random polynomials.
• Employing the truncated orthogonal ensemble of random matrices to analyze polynomial roots.
• Investigating the connection with the semi-infinite Ising spin chain model.

Main Results:

• The probability of sign persistence for the 2D diffusion field decays as p0(t) ~ t^{-θ(2)} with θ(2) = 3/16 for large t.
• The probability q0(n) that Kac's polynomials of even degree n have no real roots decays as q0(n) ~ n^{-3/4} for large n.
• A precise connection was established between these models and the Ising spin chain with Glauber dynamics.

Conclusions:

• The study provides new insights into the statistical properties of diffusion processes and random polynomials.
• The identified connections offer a unified perspective on phenomena in statistical physics and probability theory.
• The findings facilitate the computation of zero-crossing properties for diffusion fields and root properties for random polynomials.