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Stable DNA Motifs, 1D and 2D Nanostructures Constructed from Small Circular DNA Molecules

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Historical Lattice Trees.

Manuel Cabezas¹, Alexander Fribergh², Mark Holmes³

¹Pontificia Universidad Católica de Chile, Santiago, Chile.

Communications in Mathematical Physics

|June 26, 2023

Summary

This summary is machine-generated.

We show that rescaled historical processes of critical lattice trees converge to historical Brownian motion. This finding is key for understanding random walks on trees and their convergence to super-Brownian motion.

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Area of Science:

Probability Theory
Stochastic Processes
Mathematical Physics

Background:

Lattice trees are fundamental structures in statistical mechanics and probability.
Understanding the behavior of random processes on these trees is crucial for various applications.
Previous research has explored random walks and branching processes on trees, but limit theorems for historical processes were less developed.

Purpose of the Study:

To establish a functional limit theorem for rescaled historical processes associated with critical spread-out lattice trees.
To demonstrate the convergence of these processes to historical Brownian motion.
To provide a foundation for analyzing the genealogical structure of random trees.

Main Methods:

Utilizing techniques from the theory of measure-valued processes.
Applying rescaling arguments to critical spread-out lattice trees in various dimensions.
Leveraging functional limit theorems to establish convergence properties.

Main Results:

Proved that the rescaled historical processes of critical spread-out lattice trees converge to historical Brownian motion.
Established a functional limit theorem for these measure-valued processes.
The results encode the genealogical structure of the underlying random trees.

Conclusions:

The convergence to historical Brownian motion provides a powerful tool for studying random structures on trees.
These findings have direct implications for understanding the behavior of random walks on lattice trees.
The study contributes to the broader understanding of stochastic processes and their applications in mathematical physics.