Edge of spiked beta ensembles, stochastic Airy semigroups and reflected Brownian motions

The Gorin-Shkolnikov identity and its random tree generalization

“…The above theorem has the following corollary, due to [19] when x = 0 and [17] when x > 0. See also [21].…”

“…In [17], the authors show a connection between a "spiked" model similar to the model in [19] and the random variable A β . The authors compute the moment generating function for A 2 as well.…”

“…In [17], Gaudreau Lammarre and Shkolnikov extended Hariya's proof of the Gorin-Shkolnikov identity to reflected Brownian bridges conditioned on their local time at 0. If now we let…”

“…) denote a reflected Brownian bridge and denote by L = (L v t ; t ∈ [0, 1], v ≥ 0) its local time, [17,Remark 1.15] states that if (3) A β := √ 12…”

“…

In a recent pair of papers Gorin and Shkolnikov [19] and Hariya [21] have shown that the area under normalized Brownian excursion minus one half the integral of the square of its total local time is a centered normal random variable with variance 1 12 . Gaudreau Lamarre and Shkolnikov generalized this to Brownian bridges [17], and ask for a combinatorial interpretation. We provide a combinatorial interpretation using random forests on n vertices.

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