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.