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Randomized Wilson loops, reduced models and the large D expansion

Reduced models are matrix integrals believed to be related to the large N limit of gauge theories. These integrals are known to simplify further when the number of matrices D (corresponding to the number of space-time dimensions in the gauge theory) becomes large. Even though this limit appears to be of little use for computing the standard rectangular Wilson loop (which always singles out two directions out of D), a meaningful large D limit can be defined for a randomized Wilson loop (in which all D directions contribute equally). In this article, a proof-of-concept demonstration of this approach is given for the simplest reduced model (the original Eguchi-Kawai model) and the simplest randomization of the Wilson loop (Brownian sum over random walks). The resulting averaged Wilson loop displays a scale behavior strongly reminiscent of the area law.