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Post Undeleted by Ravi Andrew Bajaj
Post Deleted by Ravi Andrew Bajaj
Post Undeleted by Ravi Andrew Bajaj
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Ravi Andrew Bajaj
Ravi Andrew Bajaj
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Here is a proposal which would address your concern regarding both random trajectories in a configuration space (the complex plane Cartesian-producted with itself many times minus the space of degeneracies or collisions modded out by the symmetric group) and the statistical mechanical question of an infinite tangle (the limit as the particle number is very large for our configuration space). Consider a large matrix, say of dimension 1000, with entries all Brownian motions independent of one another. Usually the first constraint imposed on random matrices is that of symmetry or Hermiticity, but recent work (notably by Edelman and Tao and Vu and to a lesser extent Girko) examines the spectrum (or empirical spectral distribution, ESD) in the complex plane and, under certain conditions, obtains a uniform measure on the unit disk as a limit.

I haven't worked out what the SDE would be for this modification of Dyson Brownian Motion, but since the distribution on the eigenvalues still imposes a logarithmic potential (in the Gaussian case), it's likely that the property of almost surely no collisions should hold. Since we are now in the plane, the eigenvalues can move around each other as others have anticipated with prospective analysis of the statistics of the trajectories.

Here is a proposal which would address your concern regarding both random trajectories in a configuration space (the complex plane Cartesian-producted with itself many times minus the space of degeneracies or collisions modded out by the symmetric group) and the statistical mechanical question of an infinite tangle (the limit as the particle number is very large for our configuration space). Consider a large matrix, say of dimension 1000, with entries all Brownian motions independent of one another. Usually the first constraint imposed on random matrices is that of symmetry or Hermiticity, but recent work (notably by Edelman and Tao and Vu and to a lesser extent Girko) examines the spectrum (or empirical spectral distribution, ESD) in the complex plane and, under certain conditions, obtains a uniform measure on the unit disk as a limit.

I haven't worked out what the SDE would be for this modification of Dyson Brownian Motion, but since the distribution on the eigenvalues still imposes a logarithmic potential (in the Gaussian case), it's likely that the property of almost surely no collisions should hold. Since we are now in the plane, the eigenvalues can move around each other as others have anticipated with prospective analysis of the statistics of the trajectories.

Here is a proposal which would address your concern regarding both random trajectories in a configuration space (the complex plane Cartesian-producted with itself many times minus the space of degeneracies or collisions modded out by the symmetric group) and the statistical mechanical question of an infinite tangle (the limit as the particle number is very large for our configuration space). Consider a large matrix, say of dimension 1000, with entries all Brownian motions independent of one another. Usually the first constraint imposed on random matrices is that of symmetry or Hermiticity, but recent work examines the spectrum (or empirical spectral distribution, ESD) in the complex plane and, under certain conditions, obtains a uniform measure on the unit disk as a limit.

I haven't worked out what the SDE would be for this modification of Dyson Brownian Motion, but since the distribution on the eigenvalues still imposes a logarithmic potential (in the Gaussian case), it's likely that the property of almost surely no collisions should hold. Since we are now in the plane, the eigenvalues can move around each other as others have anticipated with prospective analysis of the statistics of the trajectories.

Post Deleted by Ravi Andrew Bajaj
Source Link
Ravi Andrew Bajaj
Ravi Andrew Bajaj
  • 61
  • 5

Here is a proposal which would address your concern regarding both random trajectories in a configuration space (the complex plane Cartesian-producted with itself many times minus the space of degeneracies or collisions modded out by the symmetric group) and the statistical mechanical question of an infinite tangle (the limit as the particle number is very large for our configuration space). Consider a large matrix, say of dimension 1000, with entries all Brownian motions independent of one another. Usually the first constraint imposed on random matrices is that of symmetry or Hermiticity, but recent work (notably by Edelman and Tao and Vu and to a lesser extent Girko) examines the spectrum (or empirical spectral distribution, ESD) in the complex plane and, under certain conditions, obtains a uniform measure on the unit disk as a limit.

I haven't worked out what the SDE would be for this modification of Dyson Brownian Motion, but since the distribution on the eigenvalues still imposes a logarithmic potential (in the Gaussian case), it's likely that the property of almost surely no collisions should hold. Since we are now in the plane, the eigenvalues can move around each other as others have anticipated with prospective analysis of the statistics of the trajectories.