Ostap Hryniv, Yvan Velenik
Universality of Critical Behaviour in a Class of Recurrent Random Walks
(589K, pdf)

ABSTRACT.  Let X_0=0, X_1, X_2, ..., be an aperiodic random walk 
generated by a sequence xi_1, xi_2, ..., of i.i.d. integer-valued random 
variables with common distribution p(.) having zero mean and finite 
variance. For an N-step trajectory X=(X_0,X_1,...,X_N) and 
a monotone convex function V: R^+ -> R^+ with V(0)=0, define 
V(X)= sum_{j=1}^{N-1} V(|X_j|). 
Further, let I_{N,+}^{a,b} be the set of all non-negative 
paths X compatible with the boundary conditions X_0=a, 
X_N=b. 
We discuss asymptotic properties of X in I_{N,+}^{a,b} 
w.r.t. the probability distribution 
P_{N}^{a,b}(X)= (Z_{N}^{a,b})^{-1} 
exp{-lambda V(X)} prod_{i=0}^{N-1} p(X_{i+1}-X_i) 
as N -> infinity and lambda -> 0, Z_{N}^{a,b} being the 
corresponding normalization. 
If V(.) grows not faster than polynomially at infinity, define 
H(lambda) to be the unique solution to the equation 
lambda H^2 V(H) =1. 
Our main result reads that as lambda -> 0, the 
typical height of X_{[alpha N]} scales as H(lambda) and the 
correlations along X decay exponentially on the scale 
H(lambda)^2. 
Using a suitable blocking argument, we show that the distribution 
tails of the rescaled height decay exponentially with critical 
exponent 3/2. 
In the particular case of linear potential V(.), the 
characteristic length H(lambda) is proportional to 
lambda^{-1/3} as lambda -> 0.