Unformatted text preview: n1 log P ( n1 S n ≥ a ) , a > 1 and n1 log P ( n1 S n ≤ a ) , a < 1. 4. Oriented ﬁrst passage percolation. Consider the lattice quadrant { ( i,j ) : i,j ≥ } with directed edges ( i,j ) → ( i +1 ,j ) and ( i,j ) → ( i,j +1). Associate to each edge e an exponential(1) r.v. X e , independent for diﬀerent edges. For each directed path π of length d started at (0 , 0), let S π = ∑ edges e in path X e . Let H d be the minimum of S π over all such paths π of length d . It can be shown that d1 H d → c a.s., for some constant c . Give explicit upper and lower bounds on c . [Hint: use result of previous question for lower bound.] 9...
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This note was uploaded on 11/18/2009 for the course MATH 241 taught by Professor Reed during the Spring '09 term at Duke.
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