Random Matrix Products and the Statistical Mechanics of Disordered Systems

Attention: change of lecture rooms

The course is on Wednesdays (9:00-12:00).

  • February 7th 2024: HAF478F
  • February 14th : HAF 580F
  • February 28th : Condorcet 302A
  • March 6th : HAF 580F
  • March 13th : HAF 580F
  • March 20th : HAF 580F
  • March 27th : Condorcet 302A

From the M14/RER C Bibliothèque François Mitterrand it is very simple to reach the HAF (Halle Aux Farines) building: see the map here. One has to walk toward the Seine, toward Esplanade Vidal-Naquet and the HAF is the big building on the right. It is highly advised to enter in correspondence of the big monument full of canoes. Go immediately left (again toward the Seine) and then walk up to the 4rth floor for day 1, and to the 5th in the other days (except when lectures are in Condorcet. For Condorcet go right before getting to Esplanade Vidal-Naquet (possibly it is more practical to go a bit right already when you are on Avenue de France. If you enter Condorcet from Rue Elsa Morante, go right and take stairs or elevators to the 3rd floor (if my sense of orientation is correct, the lecture room is above the crossing of rue Elsa Morante and rue Helène Brion).

Lectures:

  1. Introduction. Definition of (top) Lyapunov exponent via Fekete's Lemma. Additive co-cycles and Furstenberg-Kesten Theorem (5h. 1.5proof to be completed)
  2. End of proof (Furstenberg-Kesten). Statement and discussion of Furstenberg Theorem (Th. 1.8). Beginning of the proof: Prop. 1.9, with proof.
  3. Action of transposed matrices:Lemma 1.10, Lemma 1.13, Lemma 1.14 et Prop. 1.11 (proof almost completed). Simulations: role of “time inversion”.
  4. Completion of the proof of Furstenberg Theorem. Short introduction to Anderson localization.
  5. Anderson localization model.
  6. Completed Anderson part. Started Ising: basic facts of statistical mechanics of disordered systems. Free energy density of 1d disordered Ising model and Lyapunov exponent.
  7. Previsional: proof of the formula for the free energy. A few facts about regularity of Lyapunov exponents. Derrida-Hilhorst singularity: the heuristic argument. The weak disorder limit and the Derrida-Hilhorst singularity for the limit model.

LECTURE NOTES (updated february 13th, 2024): small changes and local corrections (all in Ch. 1)

LECTURE NOTES (updated march 20th, 2024): scattered local changes/corrections. Added proofs in App. B

LECTURE NOTES (updated march 26th, 2024): a couple of misprints, added two figures and short texts in App. B

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