Extreme Value Statistics, Integer Partitions and Bose Gas Satya N. - PowerPoint PPT Presentation

Extreme Value Statistics, Integer Partitions and Bose Gas Satya N. Majumdar Laboratoire de Physique Th´ eorique et Mod` eles Statistiques,CNRS, Universit´ e Paris-Sud, France February 27, 2008 Collaborators: A. Comtet (LPTMS, Orsay, FRANCE) P. Leboeuf (LPTMS, Orsay, FRANCE) Ref: Phys. Rev. Lett. 98, 070404 (2007) S.N. Majumdar Extreme Value Statistics, Integer Partitions and Bose Gas

Plan Plan: • A brief review on Extreme Value Statistics of i.i.d random variables ⇒ three limiting distributions: GUMBEL, FR´ = ECHET & WEIBULL S.N. Majumdar Extreme Value Statistics, Integer Partitions and Bose Gas

Plan Plan: • A brief review on Extreme Value Statistics of i.i.d random variables ⇒ three limiting distributions: GUMBEL, FR´ = ECHET & WEIBULL • Integer Partition Problem = ⇒ Ideal Bose Gas S.N. Majumdar Extreme Value Statistics, Integer Partitions and Bose Gas

Plan Plan: • A brief review on Extreme Value Statistics of i.i.d random variables ⇒ three limiting distributions: GUMBEL, FR´ = ECHET & WEIBULL • Integer Partition Problem = ⇒ Ideal Bose Gas • Same three limiting distributions in the Integer Partition/Bose gas S.N. Majumdar Extreme Value Statistics, Integer Partitions and Bose Gas

Plan Plan: • A brief review on Extreme Value Statistics of i.i.d random variables ⇒ three limiting distributions: GUMBEL, FR´ = ECHET & WEIBULL • Integer Partition Problem = ⇒ Ideal Bose Gas • Same three limiting distributions in the Integer Partition/Bose gas • Summary and Conclusions S.N. Majumdar Extreme Value Statistics, Integer Partitions and Bose Gas

Law of Averages: Central Limit Theorem • { X 1 , X 2 , . . . , X N } = ⇒ set of N i.i.d random variables —each drawn from p ( x ) → parent distribution S.N. Majumdar Extreme Value Statistics, Integer Partitions and Bose Gas

Law of Averages: Central Limit Theorem • { X 1 , X 2 , . . . , X N } = ⇒ set of N i.i.d random variables —each drawn from p ( x ) → parent distribution Average: ¯ X = X 1 + X 2 + ... + X N • N S.N. Majumdar Extreme Value Statistics, Integer Partitions and Bose Gas

Law of Averages: Central Limit Theorem • { X 1 , X 2 , . . . , X N } = ⇒ set of N i.i.d random variables —each drawn from p ( x ) → parent distribution Average: ¯ X = X 1 + X 2 + ... + X N • N • Probability distribution of ¯ X ? S.N. Majumdar Extreme Value Statistics, Integer Partitions and Bose Gas

Law of Averages: Central Limit Theorem • { X 1 , X 2 , . . . , X N } = ⇒ set of N i.i.d random variables —each drawn from p ( x ) → parent distribution Average: ¯ X = X 1 + X 2 + ... + X N • N • Probability distribution of ¯ X ? • Central Limit Theorem = ⇒ xp ( x ) dx and σ 2 = x 2 p ( x ) dx − µ 2 is finite, then for large N , if µ = � � S.N. Majumdar Extreme Value Statistics, Integer Partitions and Bose Gas

Law of Averages: Central Limit Theorem • { X 1 , X 2 , . . . , X N } = ⇒ set of N i.i.d random variables —each drawn from p ( x ) → parent distribution Average: ¯ X = X 1 + X 2 + ... + X N • N • Probability distribution of ¯ X ? • Central Limit Theorem = ⇒ xp ( x ) dx and σ 2 = x 2 p ( x ) dx − µ 2 is finite, then for large N , if µ = � � � √ � z � −∞ e − u 2 / 2 du Prob [ ¯ N ( x − µ ) 1 X ≤ x ] → G where G [ z ] = √ σ 2 π S.N. Majumdar Extreme Value Statistics, Integer Partitions and Bose Gas

Law of Averages: Central Limit Theorem • { X 1 , X 2 , . . . , X N } = ⇒ set of N i.i.d random variables —each drawn from p ( x ) → parent distribution Average: ¯ X = X 1 + X 2 + ... + X N • N • Probability distribution of ¯ X ? • Central Limit Theorem = ⇒ xp ( x ) dx and σ 2 = x 2 p ( x ) dx − µ 2 is finite, then for large N , if µ = � � � √ � z � −∞ e − u 2 / 2 du Prob [ ¯ N ( x − µ ) 1 X ≤ x ] → G where G [ z ] = √ σ 2 π 2 π e − z 2 / 2 = Prob. density: G ′ ( z ) = 1 ⇒ GAUSSIAN √ = ⇒ LAW OF AVERAGES S.N. Majumdar Extreme Value Statistics, Integer Partitions and Bose Gas

Extreme Value Statistics of i.i.d Random Variables • { X 1 , X 2 , . . . , X N } = ⇒ set of N i.i.d random variables —each drawn from p ( x ) → parent distribution S.N. Majumdar Extreme Value Statistics, Integer Partitions and Bose Gas

Extreme Value Statistics of i.i.d Random Variables • { X 1 , X 2 , . . . , X N } = ⇒ set of N i.i.d random variables —each drawn from p ( x ) → parent distribution • M N = max ( X 1 , X 2 , . . . , X N ) S.N. Majumdar Extreme Value Statistics, Integer Partitions and Bose Gas

Extreme Value Statistics of i.i.d Random Variables • { X 1 , X 2 , . . . , X N } = ⇒ set of N i.i.d random variables —each drawn from p ( x ) → parent distribution • M N = max ( X 1 , X 2 , . . . , X N ) • Q N ( x ) = Prob [ M N ≤ x ] = Prob [ X 1 ≤ x , X 2 ≤ x , . . . X N ≤ x ] S.N. Majumdar Extreme Value Statistics, Integer Partitions and Bose Gas

Extreme Value Statistics of i.i.d Random Variables • { X 1 , X 2 , . . . , X N } = ⇒ set of N i.i.d random variables —each drawn from p ( x ) → parent distribution • M N = max ( X 1 , X 2 , . . . , X N ) • Q N ( x ) = Prob [ M N ≤ x ] = Prob [ X 1 ≤ x , X 2 ≤ x , . . . X N ≤ x ] −∞ p ( x ′ ) dx ′ � N �� x � ∞ p ( x ′ ) dx ′ � N � • Independence = ⇒ Q N ( x ) = = 1 − x S.N. Majumdar Extreme Value Statistics, Integer Partitions and Bose Gas

Extreme Value Statistics of i.i.d Random Variables • { X 1 , X 2 , . . . , X N } = ⇒ set of N i.i.d random variables —each drawn from p ( x ) → parent distribution • M N = max ( X 1 , X 2 , . . . , X N ) • Q N ( x ) = Prob [ M N ≤ x ] = Prob [ X 1 ≤ x , X 2 ≤ x , . . . X N ≤ x ] −∞ p ( x ′ ) dx ′ � N �� x � ∞ p ( x ′ ) dx ′ � N � • Independence = ⇒ Q N ( x ) = = 1 − x � � x − a N • Scaling Limit: N large, x large: Q N ( x ) → F b N S.N. Majumdar Extreme Value Statistics, Integer Partitions and Bose Gas

Extreme Value Statistics of i.i.d Random Variables • { X 1 , X 2 , . . . , X N } = ⇒ set of N i.i.d random variables —each drawn from p ( x ) → parent distribution • M N = max ( X 1 , X 2 , . . . , X N ) • Q N ( x ) = Prob [ M N ≤ x ] = Prob [ X 1 ≤ x , X 2 ≤ x , . . . X N ≤ x ] −∞ p ( x ′ ) dx ′ � N �� x � ∞ p ( x ′ ) dx ′ � N � • Independence = ⇒ Q N ( x ) = = 1 − x � � x − a N • Scaling Limit: N large, x large: Q N ( x ) → F b N a N , b N → Scale factors dependent on p ( x ) F ( z ) → Scaling function: only of 3 possible varieties (depending on the tails of p ( x )) = ⇒ LAW OF EXTREMES [Fr´ echet (1927), Fisher and Tippet (1928), Gnedenko (1943), Gumbel (1958)...] → Several applications (Climate, Finance, Oceanography.....) S.N. Majumdar Extreme Value Statistics, Integer Partitions and Bose Gas

Three types of Scaling Functions: Type I (GUMBEL): If p ( x ) is unbounded with faster than power law tail (e.g., exponential) F I ( z ) = exp[ − e − z ] Type II (FR´ ECHET): If p ( x ) has power law tails: p ( x ) ∼ x − ( γ +1) F II ( z ) = 0 z ≤ 0 = exp[ − z − γ ] z ≥ 0 Type III (WEIBULL): If p ( x ) is bounded: p ( x ) ∼ (1 − x ) ( γ − 1) F III ( z ) = exp[ −| z | γ ] z ≤ 0 = 1 z ≥ 0 S.N. Majumdar Extreme Value Statistics, Integer Partitions and Bose Gas

Three types of Scaling Functions: Type I (GUMBEL): If p ( x ) is unbounded with faster than power law tail (e.g., exponential) F I ( z ) = exp[ − e − z ] Type II (FR´ ECHET): If p ( x ) has power law tails: p ( x ) ∼ x − ( γ +1) F II ( z ) = 0 z ≤ 0 = exp[ − z − γ ] z ≥ 0 Type III (WEIBULL): If p ( x ) is bounded: p ( x ) ∼ (1 − x ) ( γ − 1) F III ( z ) = exp[ −| z | γ ] z ≤ 0 = 1 z ≥ 0 1 0.8 −−−−−−−>WEIBULL 0.6 F(z) −−−−−> GUMBEL 0.4 0.2 −−−−−> FRE’CHET 0 −5 0 5 10 15 z S.N. Majumdar Extreme Value Statistics, Integer Partitions and Bose Gas

Integer Partition Problem Question: How many ways can we partition an integer E → into nonincreasing summands → Ω( E ) = ? Example: E = 4 S.N. Majumdar Extreme Value Statistics, Integer Partitions and Bose Gas

Integer Partition Problem Question: How many ways can we partition an integer E → into nonincreasing summands → Ω( E ) = ? Example: E = 4 YOUNG DIAGRAMS 4 = 4 = 3 + 1 Ω (4) = 5 = 2 + 2 = 2 + 1 + 1 = 1 + 1 + 1 +1 S.N. Majumdar Extreme Value Statistics, Integer Partitions and Bose Gas

Integer Partition Problem Question: How many ways can we partition an integer E → into nonincreasing summands → Ω( E ) = ? Example: E = 4 YOUNG DIAGRAMS 4 = 4 = 3 + 1 Ω (4) = 5 = 2 + 2 = 2 + 1 + 1 = 1 + 1 + 1 +1 Classical result of Hardy-Ramanujam (1918): for large E , � � 3 E 1 / 2 � 2 Ω( E ) ∼ exp π S.N. Majumdar Extreme Value Statistics, Integer Partitions and Bose Gas

Number of Summands in a Partition Ω( E ) → No. of partitions of E . S.N. Majumdar Extreme Value Statistics, Integer Partitions and Bose Gas

Number of Summands in a Partition Ω( E ) → No. of partitions of E . N → No. of terms (summands) in a given partition of E : N varies from one partition to another S.N. Majumdar Extreme Value Statistics, Integer Partitions and Bose Gas