First Passage Properties of the P\'olya Urn Process
Tibor Antal, E. Ben-Naim, and P.L. Krapivsky
We study first passage statistics of the P\'olya urn model. In this
random process, the urn contains two types of balls. In each step, one
ball is drawn randomly from the urn, and subsequently placed back into
the urn together with an additional ball of the same type. We derive
the probability $G_n$ that the two types of balls are equal in number,
for the first time, when there is a total of $2n$ balls. This first
passage probability decays algebraically, $G_n\sim n^{-2}$, when $n$
is large. We also derive the probability that a tie ever happens. This
probability is between zero and one, so that a tie may occur in some
realizations but not in others. The likelihood of a tie is appreciable
only if the initial difference in the number balls is of the order of
the square-root of the total number of balls.
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