Preferential Attachment When Stable
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We study an urn process with two urns, initialized with a ball each. Balls are added sequentially, the urn being chosen independently with probability proportional to the α^th power (α >1) of the existing number of balls. We study the (rare) event that the urn compositions are balanced after the addition of 2n-2 new balls. We derive precise asymptotics of the probability of this event by embedding the process in continuous time. Quite surprisingly, a fine control on this probability may be leveraged to derive a lower tail Large Deviation Principle (LDP) for L = ∑_i=1^nS_i^2/i^2, where {S_n : n ≥ 0} is a simple symmetric random walk started at zero. We provide an alternate proof of the LDP via coupling to Brownian motion, and subsequent derivation of the LDP for a continuous time analogue of L. Finally, we turn our attention back to the urn process conditioned to be balanced, and provide a functional limit law describing the trajectory of the urn process.
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