Growth-fragmentation process embedded in a planar Brownian excursion

Brownian half‐plane excursion and critical Liouville quantum gravity

“…A natural question is whether there is another natural mating of trees type theorem for 𝜅 = 4 where one has measurability in both directions. • Second, our coupling sheds light on the recent work of Aïdékon and Da Silva [1] who identify a (signed) growth fragmentation embedded naturally in the Brownian half-plane excursion. The cells in this growth fragmentation correspond to very natural observables in our exploration.…”

“…In fact, this embedded branching process was already described completely, and independently, in an earlier work of Aïdekon and Da Silva [1]. Namely, given $$X=(A,B)$$ with law as in Theorem 5.5, one can consider for any $$a\geqslant 0$$ the countable collection of excursions of $$X$$ to the right side of the vertical line with horizontal component $$a$$.…”

“… First, as stated above in Theorem 1.1, $$(\mathfrak {cle}, \mathfrak {lqg})$$ determines $$\mathfrak {be}$$, but the opposite does not hold. A natural question is whether there is another natural mating of trees type theorem for $$\kappa =4$$ where one has measurability in both directions. Second, our coupling sheds light on the recent work of Aïdékon and Da Silva [1] who identify a (signed) growth fragmentation embedded naturally in the Brownian half‐plane excursion. The cells in this growth fragmentation correspond to very natural observables in our exploration. Third, as we have already mentioned, one of the coordinates in our Brownian excursion encodes a certain LQG distance of CLE$$_4$$ loops from the boundary.…”

“…">1.Can one obtain a version of critical mating of trees where there is bi‐measurability between the decorated LQG surface and the pair of Brownian motions (with possibly additional information included)? 2.There is an interesting relation to growth‐fragmentation processes studied in [1]. Can one combine these two point of views in a fruitful way? …”

“…Associated with each such excursion is a total displacement (the difference between the vertical coordinate of the start and end points) and a sign (depending on which of these coordinates is larger). In [1], the authors prove that if one considers the evolution of these signed displacements as $$a$$ increases, then one obtains a signed growth fragmentation process with completely explicit law. The fact that this process is a growth fragmentation means, roughly speaking, that it can be described by the evolving ‘mass’ of a family of cells: the mass of the initial cell evolves according to a positive self‐similar Markov process, and every time this mass has a jump, a new cell with exactly this mass is introduced into the system.…”