Limits of Interlacing Eigenvalues in the Tridiagonal β-hermite Matrix Model

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1 Limits of Interlacing Eigenvalues in the Tridiagonal β-hermite Matrix Model Gopal K. Goel and Mentor Andrew Ahn PRIMES Conference 2017 Gopal K. Goel and Mentor Andrew Ahn About Beamer May 20, / 18

2 Matrices Recall that an m n matrix with entries in R (or C) is an array of numbers with m rows and n columns. Gopal K. Goel and Mentor Andrew Ahn About Beamer May 20, / 18

3 Matrices Recall that an m n matrix with entries in R (or C) is an array of numbers with m rows and n columns. Examples Here are examples of 3 2 and 4 4 matrices: e 1 π Gopal K. Goel and Mentor Andrew Ahn About Beamer May 20, / 18

4 Eigenvalues This is how we multiply a vector by a matrix a 11 a 12 a 13 v 1 a 11 v 1 + a 12 v 2 + a 13 v 3 a 21 a 22 a 23 v 2 = a 21 v 1 + a 22 v 2 + a 23 v 3 a 31 a 32 a 33 v 3 a 31 v 1 + a 32 v 2 + a 33 v 3 Gopal K. Goel and Mentor Andrew Ahn About Beamer May 20, / 18

5 Eigenvalues This is how we multiply a vector by a matrix a 11 a 12 a 13 v 1 a 11 v 1 + a 12 v 2 + a 13 v 3 a 21 a 22 a 23 v 2 = a 21 v 1 + a 22 v 2 + a 23 v 3 a 31 a 32 a 33 v 3 a 31 v 1 + a 32 v 2 + a 33 v 3 Examples ( ) ( ) = ( ) 3 34 Gopal K. Goel and Mentor Andrew Ahn About Beamer May 20, / 18

6 Eigenvalues We say that λ C is an eigenvalue of a square matrix A if Av = λv for some vector v. It turns out that there are n eigenvalues (up to multiplicity) of an n n matrix A. Gopal K. Goel and Mentor Andrew Ahn About Beamer May 20, / 18

7 Eigenvalues We say that λ C is an eigenvalue of a square matrix A if Av = λv for some vector v. It turns out that there are n eigenvalues (up to multiplicity) of an n n matrix A. Examples = so 3 is an eigenvalue of the original matrix. Gopal K. Goel and Mentor Andrew Ahn About Beamer May 20, / 18

8 Spectral Theorem Examples Here is a symmetric matrix: Gopal K. Goel and Mentor Andrew Ahn About Beamer May 20, / 18

9 Spectral Theorem Examples Here is a symmetric matrix: If a matrix is symmetric, then all of its eigenvalues are real. Generally, we order the eigenvalues as follows: λ 1 λ 2 λ n. Gopal K. Goel and Mentor Andrew Ahn About Beamer May 20, / 18

10 Random Variables Define a probability density p(x) to be a function such that R p(x)dx = 1. p : R R 0 Gopal K. Goel and Mentor Andrew Ahn About Beamer May 20, / 18

11 Random Variables A random variable X with values in R and density p(x) is a random number in R which can be sampled such that its frequency (histogram) as the number of samples increase converge to p(x). More precisely, Pr(a X b) = b a p(x)dx. Gopal K. Goel and Mentor Andrew Ahn About Beamer May 20, / 18

12 Example: Gaussian Random Variable A Gaussian Random Variable is one that has ( 1 p(x) = exp 2πσ 2 (x µ)2 2σ 2 Here is a sample of Gaussian random variables with µ = 0 and σ = 1. ). Gopal K. Goel and Mentor Andrew Ahn About Beamer May 20, / 18

13 Random Vectors Define a joint probability density p(x) to be a function such that R n p(x)dx n = 1. p : R n R 0 A random vector is a vector in R n that takes random values with joint distribution p(x). Gopal K. Goel and Mentor Andrew Ahn About Beamer May 20, / 18

14 Random Matrices A random matrix is a matrix whose entries are random variables. Note that the entries do not have to be independent. We can now consider the eigenvalues of these matrices, etc. Gopal K. Goel and Mentor Andrew Ahn About Beamer May 20, / 18

15 Random Matrices A random matrix is a matrix whose entries are random variables. Note that the entries do not have to be independent. We can now consider the eigenvalues of these matrices, etc. Examples PRIMES problem set problem M2! Gopal K. Goel and Mentor Andrew Ahn About Beamer May 20, / 18

16 The Model X n = 1 2 N (0, 2) χ (n 1)β χ (n 1)β N (0, 2) χ (n 2)β χ 2β N (0, 2) χ β χ β N (0, 2) It turns out that the eigenvalues have joint distribution 1 Z n 1 i<j n (λ i λ j ) β n i=1 e λ 2 i 2.. Gopal K. Goel and Mentor Andrew Ahn About Beamer May 20, / 18

17 The Model X n = 1 2 N (0, 2) χ (n 1)β χ (n 1)β N (0, 2) χ (n 2)β χ 2β N (0, 2) χ β χ β N (0, 2) It turns out that the eigenvalues have joint distribution 1 Z n 1 i<j n (λ i λ j ) β n i=1 e λ 2 i 2. Motivation. This joint distribution turns out to have an electrostatic interpretation.. Gopal K. Goel and Mentor Andrew Ahn About Beamer May 20, / 18

18 Interlacing We say that two sequences {x i } n i=1, {y j} n 1 j=1 interlace if x 1 y 1 x 2 x n 1 y n 1 x n. Gopal K. Goel and Mentor Andrew Ahn About Beamer May 20, / 18

19 Interlacing We say that two sequences {x i } n i=1, {y j} n 1 j=1 interlace if x 1 y 1 x 2 x n 1 y n 1 x n. Let A be a symmetric n by n square matrix, and let A be its n 1 by n 1 lower right submatrix. Examples A = A = ( ) Gopal K. Goel and Mentor Andrew Ahn About Beamer May 20, / 18

20 Interlacing We say that two sequences {x i } n i=1, {y j} n 1 j=1 interlace if x 1 y 1 x 2 x n 1 y n 1 x n. Let A be a symmetric n by n square matrix, and let A be its n 1 by n 1 lower right submatrix. Examples A = A = ( ) The eigenvalues of A and A interlace. Gopal K. Goel and Mentor Andrew Ahn About Beamer May 20, / 18

21 What We Are Doing With the Model Let X n 1 be the lower n 1 by n 1 submatrix of X n. We saw that the eigenvalues of X n and X n 1 interlace: Gopal K. Goel and Mentor Andrew Ahn About Beamer May 20, / 18

22 What We are Doing With the Model (continued) As n, these diagrams converge to some curve: We are interested in what this limiting shape is. Gopal K. Goel and Mentor Andrew Ahn About Beamer May 20, / 18

23 Method of Traces It turns out that the study of these diagrams is equivalent to considering what happens to trx k n trx k n 1 as n. We can work with the trace combinatorially: where trx k n trx k n 1 = i Bk k X n (i j, i j+1 ) j=1 B k = {(i 1,..., i k ) [n] k : i j i j+1 1 and i j = 1}. Gopal K. Goel and Mentor Andrew Ahn About Beamer May 20, / 18

24 Current Results Main Theorem In the β-hermite case, the diagrams converge to the Logan-Shepp curve: { 2 Ω(x) = π (x arcsin( x 2 ) + 4 x 2 ), x 2 x, x 2 We have also shown that the fluctuations of the diagrams from the curve are gaussian in some sense. Gopal K. Goel and Mentor Andrew Ahn About Beamer May 20, / 18

25 Future Work We also want to look at other random matrix models, such as β-laguerre and β-jacobi. For β-laguerre, we can describe the limiting shape, but we conjecture that the fluctuations of the diagrams from the curve are not gaussian. Gopal K. Goel and Mentor Andrew Ahn About Beamer May 20, / 18

26 Acknowledgements My mentor, Andrew Ahn Prof. Vadim Gorin for suggesting the problem Prof. Alan Edelman for useful discussions The MIT Math Department The MIT-PRIMES Program Prof. Pavel Etingof Dr. Slava Gerovitch Dr. Tanya Khovanova My Parents for supporting me throughout Gopal K. Goel and Mentor Andrew Ahn About Beamer May 20, / 18

Cylindric Young Tableaux and their Properties

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