Max-Min and Min-Max universally yield Gumbel
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"A chain is only as strong as its weakest link" says the proverb. But what about a collection of statistically identical chains: How long till all chains fail? The answer to this question is given by the Max-Min of a random matrix whose (i,j) entry is the failure time of link j of chain i: take the minimum of each row, and then the maximum of the rows' minima. The corresponding Min-Max is obtained by taking the maximum of each column, and then the minimum of the columns' maxima. The Min-Max applies to the storage of critical data. Indeed, consider multiple copies (backups) of a set of critical data items, and consider the ( i,j) matrix entry to be the time at which item j on copy i is lost; then, the Min-Max is the time at which the first critical data item is lost. In this paper we establish that the Max-Min and Min-Max of large random matrices are universally governed by asymptotic Gumbel statistics. We further establish that the domains of attraction of the asymptotic Gumbel statistics are effectively all-encompassing. Also, we show how the asymptotic Gumbel statistics can be applied to the design of large systems.
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