Here's a (hopefully clearer) statement of the question. Consider a random walk on ZxZ that starts at (0,0) and progresses n-1 steps upward and rightward, ending at some vertex (i,j) with i,j non-negative integers summing to n-1, such that p_{i,j} is the probability that a walker at (i,j) will go to (i+1,j) (and 1-p_{i,j} is the probability that a walker at (i,j) will go to (i,j+1)). Thus the path taken in the first n-1 steps is determined by all the p_{i,j}'s with i,j non-negative integers with sum no greater than n-2. Suppose we want to choose those n(n-1)/2 probabilities p_{i,j} so that the probability of the walker being at (n-k,k-1) is some specified number p_k (1 leq k leq n). This is a massively underdetermined problem; there are lots of ways to choose biases at the n(n-1)/2 junctions so as to generate this distribution after n steps. But suppose we have the additional goal of minimizing the effect that slight perturbations of the biases will have. Is there a unique best choice of biases, and if so, what is it? Here we're dealing with a map from R^{n(n-1)/2} to R^n, so it's not clear to me what the right way to measure sensitivity to perturbations is. But I suspect that this problem has a nice canonical answer if one chooses some natural way to measure sensitivity. What I actually want is not necessarily the most noise-insensitive way of biasing the junctions, but a reasonably noise-insensitive way of biasing the junctions that is also easy to compute and mathematically natural. (Yes, "natural" is even more vague than "noise-insensitive".) When I posted the problem I thought that minimizing noise-sensitivity would lead to a nice way of biasing the junctions, but now I'm not so sure, since I've done some more analysis of my original idea about how to measure noise-sensitivity, and it leads to nasty algebraic expressions (e.g., polynomial equations that can't be solved in terms of radicals). Jim Propp