Hammersley ’ s Path Process 2
نویسنده
چکیده
where each ( ) ∈ , 1 ≤ ≤ , and is arbitrary. For convenience, define (0 0) = (0 0) and (+1 +1) = (1 1). Define such a point sequence to be an up/right path if, for any ≥ 1, we have −1 ≤ and −1 ≤ . Hence an up/right path joins points of in a continuous, piecewise linear manner with line segments of slope , 0 ≤ ≤ ∞, attaching (−1 −1) and ( ) for all . Of all up/right paths determined by , there is (at least) one with a maximum number of points. Call this number . (This is usually referred to as a length in the literature. Of course, it also depends implicitly on + and −.) What can be said about the probability distribution of as →∞? A special case of the above is the longest increasing subsequence problem [4], achieved when + = − = 0. Its solution will be folded into the formulas we give shortly for the general problem. This turns out to be related to the polynuclear growth (PNG) model in physics due to Prähofer & Spohn [5, 6, 7], but we cannot discuss such topics now.
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Beta-paths in the Hammersley Process
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