[Sur le volume de l'intersection d'une boule avec des demi espaces aléatoires]
Nous trouvons une expression asymptotique du volume de l'intersection d'une boule à N dimensions avec p=αN demi espaces aléatoires quand α ne depasse pas la valeur critique αc. Cette expression est la même que celle trouvée par Gardner [3] en utilisant un calcul de repliques. Nous trouvons aussi la mème valeur de αc. Notre démonstration est rigoureuse et basée sur la methode de la cavité. La nécessaire décroissance des corrélations est obtenue en utilisant un argument géométrique qui est vrai pour des hamiltoniens généraux.
We find an asymptotic expression of the volume of the intersection of the N dimensional sphere with p=αN random half spaces when α is less than a critical value. This expression coincides with the one found by Gardner [3] using replica calculations. We get also the same value for αc. Our proof is rigorous and based on the cavity method. The required decay of correlations is obtained by means of a geometrical argument which holds for general Hamiltonians.
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@article{CRMATH_2002__334_9_803_0,
author = {Shcherbina, Maria and Tirozzi, Brunello},
title = {On the volume of the intersection of a sphere with random half spaces},
journal = {Comptes Rendus. Math\'ematique},
pages = {803--806},
publisher = {Elsevier},
volume = {334},
number = {9},
year = {2002},
doi = {10.1016/S1631-073X(02)02345-2},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/S1631-073X(02)02345-2/}
}
TY - JOUR AU - Shcherbina, Maria AU - Tirozzi, Brunello TI - On the volume of the intersection of a sphere with random half spaces JO - Comptes Rendus. Mathématique PY - 2002 SP - 803 EP - 806 VL - 334 IS - 9 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/S1631-073X(02)02345-2/ DO - 10.1016/S1631-073X(02)02345-2 LA - en ID - CRMATH_2002__334_9_803_0 ER -
%0 Journal Article %A Shcherbina, Maria %A Tirozzi, Brunello %T On the volume of the intersection of a sphere with random half spaces %J Comptes Rendus. Mathématique %D 2002 %P 803-806 %V 334 %N 9 %I Elsevier %U http://archive.numdam.org/articles/10.1016/S1631-073X(02)02345-2/ %R 10.1016/S1631-073X(02)02345-2 %G en %F CRMATH_2002__334_9_803_0
Shcherbina, Maria; Tirozzi, Brunello. On the volume of the intersection of a sphere with random half spaces. Comptes Rendus. Mathématique, Tome 334 (2002) no. 9, pp. 803-806. doi : 10.1016/S1631-073X(02)02345-2. http://archive.numdam.org/articles/10.1016/S1631-073X(02)02345-2/
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