Embedded trees and the support of the ISE

Area of Science:

• Probability Theory
• Random Trees
• Mathematical Physics

Background:

• Embedded trees are rooted trees with specific label constraints on adjacent vertices.
• Previous work linked maximum/minimum labels in these trees to the support of integrated superbrownian excursion (ISE).

Purpose of the Study:

• To explicitly detail the probabilistic limiting relation between embedded trees and ISE.
• To utilize generating functions and Jacobi's theta functions for this analysis.

Main Methods:

• Employing a generating function approach based on Bouttier et al. (2003).
• Utilizing properties of Jacobi's theta functions.
• Deriving integral representations involving the Weierstrass P-function.

Main Results:

• An integral representation for the joint distribution of the supremum and infimum of the ISE support was derived.
• The limiting radius distribution in random quadrangulations was re-derived using exact counting generating functions.

Conclusions:

• The study provides a more explicit probabilistic connection between embedded trees and ISE.
• The methods offer new insights into the structure of random trees and related stochastic processes.