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Jonas Arista1, Neil O'Connell1

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This study explores connections between random matrices and non-intersecting processes using Fomin

Keywords:
Loop-erased walksNon-intersecting Brownian motionsRandom matrix theorySLE

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Area of Science:

  • Probability Theory
  • Mathematical Physics
  • Stochastic Processes

Background:

  • Non-intersecting processes in one dimension are linked to random matrices via the reflection principle.
  • Fomin generalized the reflection principle for planar processes, involving loop-erased paths instead of non-intersection.
  • Sato and Katori (2011) observed connections between independent Brownian motions and random matrices in planar domains.

Purpose of the Study:

  • To present new examples of random matrix ensembles arising from Brownian motions.
  • To extend Fomin's identity to the affine setting.
  • To provide a novel interpretation of the circular orthogonal ensemble.

Main Methods:

  • Utilizing Fomin's generalized reflection principle.
  • Analyzing independent Brownian motions in planar domains and annuli.
  • Deriving Cauchy-type ensembles and the circular orthogonal ensemble.

Main Results:

  • Identified new examples of Cauchy-type ensembles.
  • Extended Fomin's identity to the affine setting.
  • Established a novel interpretation of the circular orthogonal ensemble using Brownian motions in an annulus.

Conclusions:

  • The study reveals deeper connections between stochastic processes and random matrix theory.
  • Fomin's identity and Brownian motion provide a unified framework for certain random matrix ensembles.
  • This work offers new perspectives on the structure and interpretation of specific random matrix types.