On the evolution of islands |
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Authors: | Peter G Doyle Colin Mallows Alon Orlitsky Larry Shepp |
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Institution: | (1) AT&T Bell Laboratories, 07974 Murray Hill, NJ, USA;(2) Princeton University, 08544 Princeton, NJ, USA |
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Abstract: | Letn cells be arranged in a ring, or alternatively, in a row. Initially, all cells are unmarked. Sequentially, one of the unmarked
cells is chosen at random and marked until, aftern steps, each cell is marked. After thekth cell has been marked the configuration of marked cells defines some number of islands: maximal sets of adjacent marked
cells. Let ξ
k
denote the random number of islands afterk cells have been marked.
We give explicit expressions for moments of products of ξ
k
’s and for moments of products of 1/ξ
k
’s.
These are used in a companion paper to prove that if a random graph on the natural number is made by drawing an edge betweeni≧1 andj>i with probabilityλ/j, then the graph is almost surely connected ifλ>1/4 and almost surely disconnected ifλ≦1/4. |
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Keywords: | |
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