Scaling limit under conditioning on the size of the underlying tree

Determine the explicit distributional limit of the subordinate tree associated with a centered random snake indexed by a critical Bienaymé tree conditioned to have n edges, including the limiting behavior of its size and its dependence on the conditioned underlying tree.

Background

The paper establishes joint convergence of the underlying conditioned Bienaymé tree, its branching random walk, and a contour encoding of the subordinate tree. However, when the underlying tree is conditioned to have a fixed number n of edges, the subordinate tree is not itself a Bienaymé tree under an intrinsic conditioning event, because the conditioning depends on the original tree’s size.

The unresolved issue is to identify the explicit limiting law of the subordinate tree in this size-conditioned setting. The authors indicate that understanding the distributional limit of the number of records, or equivalently the size of the subordinate tree, may be a route toward such a description; they only prove an asymptotic formula for its expectation.

References

However, the dependence between the size of $T_n$ and the shape of $S(W_n)$ is still unclear thus we were not able to give an explicit description of the limit law of $S(W_n)$. Nonetheless, we believe that understanding the distributional limit of $#S(W_n)$ could lead to this explicit description, and we provide its asymptotic in expectation.

— Subordination of discrete snakes  (2610.02115 - Aurillard et al., 1 Oct 2026) in Section 2, subsection “Scaling limits of large BRW and their subordinate trees,” paragraph immediately preceding Theorem~\ref{thm:expectedNumberOfRecords}