An Exponential Limit Shape of Random q-proportion Bulgarian Solitaire

• We introduce pn-random qn-proportion Bulgarian solitaire (0 < pn, qn ≤ 1), playedon n cards distributed in piles. In each pile, a number of cards equal to the propor-tion qn of the pile size rounded upward to the nearest integer are candidates to bepicked. Each candidate card is picked with probability pn, independently of othercandidate cards. This generalizes Popov’s random Bulgarian solitaire, in whichthere is a single candidate card in each pile. Popov showed that a triangular limitshape is obtained for a fixed p as n tends to infinity. Here we let both pn and qnvary with n. We show that under the conditions q2npnn/log n → ∞ and pnqn → 0 asn → ∞, the pn-random qn-proportion Bulgarian solitaire has an exponential limitshape.

Ämnesord

NATURVETENSKAP  -- Matematik -- Sannolikhetsteori och statistik (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Probability Theory and Statistics (hsv//eng)

Nyckelord

Mathematics/Applied Mathematics

matematik/tillämpad matematik

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