Erich's second idea (if I understand it) won't work for the case of the probability vector (1/2,0,1/2). For this vector, we really have no choice about what biases to use: we must use bias 1/2:1/2 at the top of the quincunx, bias 1:0 in the second row at the left, and bias 0:1 in the second row at the right. (Here I'm reverting to the standard orientation of the quincunx; just take the first-quadrant picture from my last email and rotate it 45 degrees counterclockwise.) Note that the biases 1:0 and 0:1 are different. A variant that Erich's posting suggested to me is the idea of having p_{i,j} depend only on i-j; I'll have to think about it. In the meantime, though, I want to point out that the degrees-of-freedom count is off. As (i,j) varies over all locations where a routing-junction exists, i-j takes on all values between 0-(n-2) and (n-2)-0; so there are 2(n-2)+1 = 2n-3 variables to play with, but there are only n constraints (or maybe just n-1, since p_1,p_2,...,p_n must sum to 1). This discrepancy doesn't appear in the case n=2 (where there's just one routing-junction), but it appears in the case n=3. Jim Propp On Mon, Jun 18, 2012 at 12:54 PM, Erich Friedman <efriedma@stetson.edu>wrote:
you need a precise definition of how to measure "the effect [of] slight perturbations".
a natural definition would be to measure the maximum value of | p_k - P |, where p_k is the probability of passing through (n-k,k-1) and P are all the probabilities of passing through that same point where the individual steps have probability p_{i,j} + - epsilon instead of p_{i,j}. but this seems very hard, and is not likely to give nice results.
a slightly different approach: since there are n degrees of freedom here, and the random walk takes n steps, is it possible to make all the p_{i,j} equal if i+j has constant sum (in other words, each step's transition probabilities are independent of the path so far) ?
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