The reunions of three dissimilar vicious walkers

Abstract

We study the behavior of three “vicious” random walkers which diffuse freely in one dimension witharbitrary diffusivitiesb ²₁ ,b ²₂ ,b ²₃ , except that their paths may not cross. The full distribution function is calculated exactly in the continuum limit; the exponent ψ₃ governing the decay of the probability$R_n^{(3)} \sim 1/n^{\psi _3 }$ of a simultaneousreunion of all three walkers aftern steps is found to varycontinuously according to$\psi _3 = 1 + \pi /\cos ^{ - 1} \left\{ {b_2^2 /\left[ {(b_1^2 + b_2^2 )(b_2^2 + b_3^2 )} \right]^{1/2} } \right\}$. This variation has consequences for various interfacial wetting transitions in (1+1) dimensions. It may also be related heuristically to the marginality of direct interface-wall interactions decaying asW ₀/l ² in the intermediate fluctuation regime of (1+1)-dimensional wetting, where exponents varying continuously withW ₀ have recently been found.