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The diffusion equation ₓ0ex{0ex}=0ex{0ex}▽^2 is considered, with initial condition (x, 0), a Gaussian random variable with zero mean. Using a simple approximate theory we show that the probability p₍ (t₁, t₂) that (x, t) (for a given space point x) changes sign n times between t₁ and t₂ has the asymptotic form p₍ (t₁, t₂) c₍ln (t₂/t₁) ^n (t₁/t₂) ^-. The exponent has predicted values 0. 1203, 0. 1862, 0. 2358 in dimensions d0ex{0ex}=0ex{0ex}1, 2, 3, in remarkably good agreement with simulation results.
Majumdar et al. (Mon,) studied this question.