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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.