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Decomposition of densities in individual contributions

Decomposition of densities in individual contributions. M. P. Pato University of São Paulo (USP). Tracy-Widom distributions. Applications: Longest increasing subsequence in a random permutation follows F 2, and also F 1 and F 4 Growth processes F 2 and also F 1 Random tilings F 2

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Decomposition of densities in individual contributions

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  1. Decomposition of densities in individual contributions M. P. Pato University of São Paulo (USP)

  2. Tracy-Widom distributions

  3. Applications: • Longest increasing subsequence in a random permutation follows F2, and also F1 and F4 • Growth processes F2and also F1 • Random tilings F2 • Queuing theory F2 • Universality: • TW hold if is replaced by • Impact: it is a distribution of extreme values of correlated sequences

  4. Poisson process • For a i.i.d. sequence of density ρ(x) the probability of • the extreme value xmax be less than a value t is • F ~ exp [-exp (-y) ], (Gumbel) if ρ(x) decays fast (exponentially) • F ~ exp( - 1/yμ) , (Fréchet) if ρ(x) decays with power μ+1 • F ~ exp( y) , (Weibull) if ρ(x) is bounded • y is properly normalized • FN → F max-stability property → universality

  5. Tracy-Widom distributions

  6. Poisson process • For a i.i.d. sequence of density ρ(x) the probability of • the extreme value xmax be less than a value t is • F ~ exp [-exp (-y) ], (Gumbel) if ρ(x) decays fast (exponentially) • F ~ exp( - 1/yμ) , (Fréchet) if ρ(x) decays with power μ+1 • F ~ exp( y) , (Weibull) if ρ(x) is bounded • y is properly normalized • FN → F max-stability property → universality

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