Tails in harnesses

Abstract

We consider space- and time-uniformd-dimensional random processes with linear local interaction, which we call harnesses and which may be used as discrete mathematical models of random interfaces. Their components are rea random variablesa ^t_s , wheres ∈ Z^d andt=0, 1, 2.,... At every time step two events occur: first, every component turns into a linear combination of itsN neighbors, and second, a symmetric random i.i.d. “noise”v is added to every component. For any σ ∈Z ⁺_d define Δ_σ a′ _s as follows. If σ=(0,...,0), σ=(0,...,0), Δ_σ a ^t _s =a ^t _s . Then by induction,$\Delta _{\sigma + e_1 } a_s^t = \Delta _\sigma a_{s + e_1 }^t - \Delta _\sigma a_s^t$ wheree _i is thed-dimensional vector, whoseith component is one and other components are zeros. Denote |σ| the sum of components of σ. Call a real random variable ϕ symmetric if it is distributed as −ϕ. For any symmetric random variable ϕpower decay or P-decay is defined as the supremum of thoser for which therth absolute moment of ϕ is finite. Convergence a.s., in probability and in law whent→∞ is examined in terms of P-decay(v): Ifd=1, σ=0 ord=2, σ=(0,0), Δ_σ a ^t _s diverges. In all the other cases: If P-decay(v)<(d+2)/(d+|σ|), Δ_σ a ^t _s diverges; if P-decay(v)>(d+2)/(d+|σ|), Δ_σ a ^t _s , converges and P-decay(ν) For any symmetric random variable ϕexponential decay or E-decay is defined as the supremum of thoser for which the expectation of exp(|x|^r) is finite. Let E-decay(v)>0. Whenever Δ_σ a ^t _s converges (that is, ifd>2 or |σ|>0: Ifd>2, E-decay(lima ^t_s )=min(E-decay(v),d+2/2); if |σ|=1, E-decay (lim Δ_σ a ^t _s )=min(E-decay(ν),d+2); if |σ| ⩾, E-decay (lim Δ_σ a ^t _s )=E-decay(ν).