On the triviality of the shocked map

TL;DR

This paper investigates the metric properties of tree-decorated quadrangulations in the critical regime, showing they converge to a Brownian disk with a point-identified boundary, implying the triviality of the shocked map.

Contribution

It establishes the convergence of critical tree-decorated quadrangulations to a Brownian disk with boundary identification, revealing the triviality of the shocked map.

Findings

Diameter of the tree scales as f^{1/4} with logarithmic corrections.

Critical quadrangulations converge to a Brownian disk after rescaling.

The shocked map is shown to be trivial as a metric space.

Abstract

The (non-spanning) tree-decorated quadrangulation is a random pair formed by a quadrangulation and a subtree chosen uniformly over the set of pairs with prescribed size. In this paper we study the tree-decorated quadrangulation in the critical regime: when the number of faces of the map, $f$, is proportional to the square of the size of the tree. We show that with high probability in this regime, the diameter of the tree is between $o(f^{1/4})$ and $f^{1/4}/\log^\alpha(f)$, for $\alpha >1$. Thus after scaling the distances by $f^{-1/4}$, the critical tree-decorated quadrangulation converges to a Brownian disk where the boundary has been identified to a point. These results imply the triviality of the shocked map: the metric space generated by gluing a Brownian disk with a continuous random tree.