Limit Theorems for Jacobi Ensembles

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1 Limit Theorems for Jacobi Esembles Chiheo Kim Departmet of Mathematics MIT May 5th, 203 Chiheo Kim Limit Theorems for Jacobi Esembles May 5th, 203 / 9

2 Jacobi Esembles A set of radom variables is a (complex, β = 2) Jacobi esemble if that ca be described i the followig equivalet ways: for parameters (, m, m 2 ), (λ,..., λ ) [0, ] with joit pdf C,m,m 2 i<j λ i λ j 2 i= λ m i ( λ i ) m 2. Eigevalues of matrix X = AA /(AA + BB ) where A, B are matrices of size m ad m 2 with i.i.d. stadard (complex) Gaussia radom variables. Equivaletly, it is the squared geeralized sigular values of the pair A, B. Chiheo Kim Limit Theorems for Jacobi Esembles May 5th, / 9

3 Jacobi Esembles A set of radom variables is a (complex, β = 2) Jacobi esemble if that ca be described i the followig equivalet ways: for parameters (, m, m 2 ), Eigevalues of a radom matrix of the form π ππ where π ad π are idepedet (m + m 2 ) (m + m 2 ) radom orthogoal projectios of raks ad m, whose distributios are ivariat uder uitary cojugatio. Eigevalues of U U where U is the m upper-left corer of a (m + m 2 ) (m + m 2 ) Haar (uitary) matrix. Log-gas model, etc. Chiheo Kim Limit Theorems for Jacobi Esembles May 5th, / 9

4 Jacobi Esembles Ca geeralize to ay β > 0. CS values of certai bidiagoal matrix. (Edelma & Sutto 2003) Eigevalues of certai tridiagoal matrix. (Killip & Neciu 2004) Chiheo Kim Limit Theorems for Jacobi Esembles May 5th, / 9

5 Jacobi Esembles Let c s s c B β = c s 2 s 2 c 2 c 2 s s c c with ad s i = c 2 i Beta( β 2 (m + i), β 2 (m 2 + i)) c j 2 Beta( β 2 j, β 2 (m + m j)), c 2i =, s j = c j 2. The, the eigevalues of B β B T β has same distributio as Jacobi esemble. Chiheo Kim Limit Theorems for Jacobi Esembles May 5th, / 9

6 Limit theorems If we sample values from a certai distributio, the how would the average look like? Idepedet radom variables: Law of large umbers. X X µ Hermite esembles (GOE, GUE,... ) : Wiger s semicircle law δ λi 2 x π 2 dx i= Laguerre esembles : Marcheko-Pastur law (x γ )(γ + x) δ λi [γ,γ +] dx 2πγx i= Chiheo Kim Limit Theorems for Jacobi Esembles May 5th, / 9

7 Limit theorems How about Jacobi esembles? Let parameters be (, m, m 2 ).. If m + m 2 2 = o(), the i= δ λi π x( x) dx. 2. If m / p ad m 2 / q ad p + q > 2, the δ λi p + q (λ+ x)(x λ ) 2π x( x) [λ,λ +]dx, where i= ( p λ ± = p + q ( p + q ) ± p + q ( ) 2 p p + q ). Chiheo Kim Limit Theorems for Jacobi Esembles May 5th, / 9

8 Limit theorems How about Jacobi esembles? Let parameters be (, m, m 2 ). 3. If m + m 2 2 = ω() ad if m m +m 2 2 λ, the δ λi δ λ. i= (Every covergece here is i probability.) We ca say more about the third case if we scale appropriately. Chiheo Kim Limit Theorems for Jacobi Esembles May 5th, / 9

9 Limit theorems Theorem Assume that β 2m γ (0, ], ad m 2 = ω( 2 ). The, i= δ m 2 λ i c f γ (cx)dx weakly, where c = 2γ/β ad f γ is the desity fuctio of Marcheko-Pastur law with parameter γ, i.e., f γ (x) = (x γ )(γ + x) 2πγx [γ+,γ ] ad γ ± = ( γ ± ) 2. Chiheo Kim Limit Theorems for Jacobi Esembles May 5th, / 9

10 Fluctuatios Oe may thik about the deviatio. For example, Radom variables: Cetral limit theorem. β-hermite esemble: Arcsi law. For ay polyomial φ, ( ) 2 φ(λ i ) = φ(x)dσ(x) + β φ(x)dµ H (x) + o(/), i= where σ has semicircle distributio, ad dµ H = 4 δ + 4 δ β-laguerre esemble: Similar, we have dµ L = 4 δ γ + 4 δ γ + dx 2π x 2. dx 2π (x γ )(γ + x). Chiheo Kim Limit Theorems for Jacobi Esembles May 5th, / 9

11 Fluctuatios I the case of truly Jacobi esemble, we ca calculate the deviatio. Theorem If m / p ad m 2 / q, the for ay polyomial φ, i= λ+ ( ) 2 φ(λ i ) = φ(x)dµ(x) + λ β φ(x)dµ J (x) + o(/), where µ is the limitig distributio, ad dµ J = 4 δ λ + 4 δ dx λ + 2π (x λ )(λ + x) Chiheo Kim Limit Theorems for Jacobi Esembles May 5th, / 9

12 Refereces I. Dumitriu ad A. Edelma. Global spectrum uctuatios for the β-hermite ad β-laguerre esembles via matrix models. J. Math. Phys. 47 (2006), o. 6, , 36. MR (2007e:8204) 2 T. Jiag. Approximatio of Haar Distributed Matrices ad Limitig Distributios of Eigevalues of Jacobi Esembles. Probability Theory ad Related Fields, 44() (2009), I. Dumitriu ad E. Paquette. Global Fluctuatios for Liear Statistics of β-jacobi Esemble. Radom Matrices: Theory Appl. 0 (202), Chiheo Kim Limit Theorems for Jacobi Esembles May 5th, / 9