Decomposition of densities in individual contributions

M. P. Pato

University of São Paulo (USP)

Tracy-Widom distributions

Applications:

1. Longest increasing subsequence in a random permutation follows F2, and also F1 and F4

2. Growth processes F2 and also F1

3. Random tilings F2

4. Queuing theory F2

Universality:

TW hold if is replaced by

Impact: it is a distribution of extreme values of correlated sequences

2tr H HVtr

Poisson process

• 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

t

N

tdxxdxx

NtxF ])(exp[)(

11)( max

For a i.i.d. sequence of density ρ(x) the probability of

the extreme value xmax be less than a value t is

Tracy-Widom distributions

Poisson process

• 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

t

N

tdxxdxx

NtxF ])(exp[)(

11)( max

For a i.i.d. sequence of density ρ(x) the probability of

the extreme value xmax be less than a value t is