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Mutually avoiding paths in random media and largest eigenvalues of random matrices.

Andrea De Luca1, Pierre Le Doussal2

  • 1Laboratoire de Physique Théorique et Modèles Statistiques (UMR CNRS 8626), Université Paris-Sud, Orsay, France.

Physical Review. E
|April 19, 2017
PubMed
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The study shows that the free energies of multiple directed polymers (DP) in random potentials align with sums of random matrix eigenvalues. This confirms a conjecture about directed polymers and random matrix theory for N>1 noncrossing polymers.

Area of Science:

  • Statistical mechanics
  • Random matrix theory
  • Condensed matter physics

Background:

  • The free energy distribution of directed polymers (DP) in random potentials, akin to the Kardar-Parisi-Zhang (KPZ) equation, converges to the Tracy-Widom distribution.
  • The Tracy-Widom distribution characterizes largest eigenvalue fluctuations in Gaussian unitary ensembles (GUE).
  • This convergence was previously established for single DPs with fixed endpoints (droplet initial conditions).

Purpose of the Study:

  • To test a conjecture extending the Tracy-Widom distribution to multiple noncrossing directed polymers (N>1).
  • To investigate the relationship between the free energies of N>1 noncrossing continuum DPs and sums of Nth largest GUE eigenvalues.
  • To provide an exact calculation of the right tails of these probability distribution functions (PDFs).

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Main Methods:

  • Utilizing replica methods to analyze the statistical properties of the system.
  • Calculating the exact right tails of the probability distribution functions (PDFs) for free energies.
  • Comparing the calculated PDFs with predictions from random matrix theory.

Main Results:

  • The study provides an exact calculation of the right tails of the PDFs for the free energies of N>1 noncrossing continuum DPs.
  • The calculated right tails of the PDFs were shown to coincide with the sums of the Nth largest eigenvalues of the GUE for arbitrary N.
  • This demonstrates a precise agreement between the statistical properties of directed polymers and random matrix theory predictions.

Conclusions:

  • The findings offer strong evidence supporting the conjecture relating free energies of multiple noncrossing DPs to sums of GUE eigenvalues.
  • The results extend the known connection between directed polymers and random matrix theory beyond the single polymer case.
  • This work deepens the understanding of universal fluctuations in disordered systems and their connection to random matrix theory.