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Integrable matrix probabilistic diffusions and the matrix stochastic heat equation.

Alexandre Krajenbrink1, Pierre Le Doussal2

  • 1Quantinuum, Partnership House, Carlisle Place, London SW1P 1BX, United Kingdom and Le Lab Quantique, 58 rue d'Hauteville, 75010 Paris, France.

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Summary
This summary is machine-generated.

We introduce a matrix stochastic heat equation (MSHE) and find its invariant measure. This integrable model allows studying large deviations via inverse scattering, connecting to matrix polymers.

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Area of Science:

  • Mathematical Physics
  • Statistical Mechanics
  • Stochastic Processes

Background:

  • The study of stochastic partial differential equations is crucial for modeling complex systems.
  • Integrable systems offer powerful analytical tools for understanding dynamics.
  • Matrix-valued stochastic processes are increasingly important in diverse fields.

Purpose of the Study:

  • Introduce and analyze a matrix version of the stochastic heat equation (MSHE).
  • Determine the explicit invariant measure for the MSHE in one spatial dimension.
  • Investigate the integrability and large deviation properties of the MSHE and related discrete models.

Main Methods:

  • Derivation of the invariant measure for the MSHE.
  • Demonstration of classical integrability in the weak-noise regime using matrix nonlinear Schrödinger equation.
  • Application of inverse scattering techniques for short-time large deviation analysis.
  • Analysis of discrete matrix polymer models, including matrix log-Gamma and O'Connell-Yor polymers.
  • Utilizing fluctuation-dissipation transformations on dynamical actions.

Main Results:

  • Explicit invariant measure obtained for the 1D MSHE.
  • Classical integrability shown for the MSHE in the weak-noise limit.
  • MSHE identified as a continuum limit of the matrix log-Gamma polymer.
  • Classical integrability confirmed for discrete matrix polymer models.
  • Lax pairs and invariant measures derived for all studied models.

Conclusions:

  • The MSHE and related discrete models exhibit classical integrability.
  • The developed methods provide a framework for analyzing large deviations in these systems.
  • Connections between continuum stochastic equations and discrete polymer models are established.