The Gorin-Shkolnikov identity and its random tree generalization

In a recent pair of papers Gorin and Shkolnikov (2018) and Hariya (2016) 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 $\frac{1}{12}$. Gaudreau Lamarre and Shkolnikov (2019) generalized this to Brownian bridges, and ask for a combinatorial interpretation. We provide a combinatorial interpretation using random forests on $n$ vertices. In particular, we show that there is a process level generalization for a certain infinite forest model. We also show analogous results for a variety of other related models using stochastic calculus.