 03469 Ostap Hryniv, Yvan Velenik
 Universality of Critical Behaviour in a Class of Recurrent Random Walks
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Oct 15, 03

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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. integervalued random
variables with common distribution p(.) having zero mean and finite
variance. For an Nstep 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}^{N1} V(X_j).
Further, let I_{N,+}^{a,b} be the set of all nonnegative
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}^{N1} 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.
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