Convexity and Intersection of Random Spaces
The problem of finding the volume of the intersection of the N dimensional sphere with p = αN random half spaces when α is less than a critical value α c , and when N,p → ∞is solved rigorously. The asymptotic expression coincides with the one found by E. Gardner (), using non rigorous replica calculations in neural network theory. When α is larger than α c the volume of the intersection goes to 0 more rapidly than exp(— N const). We use the cavity method. The convexity of the volume and the Brunn Minkowski theorem () have a central role in the proof.
KeywordsRandom systems Convexity
Unable to display preview. Download preview PDF.
- Bovier A., Gayrard V., Hopfield Models as a Generalized Random Mean Field Models, Mathematical Aspects of Spin Glasses and Neuronal Networks, A. Bovier, P. Picco Eds., Progress in Probability, Birkhauser, 41 (1998), 3–89.Google Scholar
- Hadwiger H., Vorlesungen uber Inhalt, Oberlache und Isoperimetrie. Springer-Verlag, 1957.Google Scholar
- Ghirlanda S., Guerra F., General Properties of Overlap Probability Distributions in Disordered Spin System, J.Phys.A: Math.Gen. 31(1988)9149–9155.Google Scholar
- Shcherbina M., Some Estimates for the Critical Temperature of The Sherrington-Kirkpatrick Model with Magnetic Field, Mathematical Results in Statistical Mechanics World Scientific, Singapore (1999) 455–474.Google Scholar
- Shcherbina M., Tirozzi, B., Rigorous Solution of the Gardner Problem, to appear.Google Scholar
- Talagrand M., Rigorous Results for the Hopfield Model with Many Patterns. Prob. Theor. Rel. Fields 110 (1998) 176–277.Google Scholar