Correlations in Uniform Spanning Trees: a Fermionic Approach
1Utrecht University, Budapestlaan 6, 3584 CD Utrecht, The Netherlands.
Journal of Statistical Physics
|October 21, 2025
Summary
We found a direct link between uniform spanning tree edge probabilities and fermionic Gaussian free field states. This connection allows for explicit calculations of tree degrees and their scaling limits.
Area of Science:
- Probability Theory
- Statistical Mechanics
- Graph Theory
Background:
- Uniform Spanning Trees (UST) are fundamental objects in graph theory.
- Fermionic Gaussian Free Fields (FGFF) are important in statistical physics and random matrix theory.
- Understanding the relationship between these two areas can reveal deeper structural properties.
Purpose of the Study:
- To establish a precise correspondence between UST edge probabilities and FGFF states.
- To leverage this connection for explicit calculations involving USTs.
- To analyze the degree distribution of USTs and their asymptotic behavior.
Main Methods:
- Expressing UST edge probabilities using fermionic Gaussian expectations.
- Developing methods for explicit calculation of joint probability mass functions.
- Investigating scaling limits on regular lattices.
Main Results:
- A clear mathematical correspondence is established between UST edge probabilities and FGFF states.
- Explicit formulas are derived for the joint probability mass functions of UST degrees.
- Scaling limits for UST degrees on certain regular lattices are obtained.
Conclusions:
- The established link provides a powerful new tool for studying USTs.
- The methods allow for exact computations and asymptotic analysis of UST properties.
- This work bridges concepts from graph theory and advanced statistical physics.
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