Random matrix theory, numerical computation and applications

Abstract

Abstract. This paper serves to prove the thesis that a computational trick can open entirely new approaches to theory. We illustrate this by describing such random matrix techniques as the stochastic operator approach, the method of ghosts and shadows, and the method of "Riccatti Diffusion/Sturm Sequences." We thereby provide new insights into the deeper mathematics underlying random matrix theory.

Numerical Simulation of GUE Two-Point Correlation and Cluster Functions

“…The Gaussian orthogonal ensemble (GOE), Gaussian unitary ensemble (GUE) and Gaussian symplectic ensemble (GSE) are further defined by three possible symmetries of H which result in real eigenvalues [3,22]. In the case of the GUE, a member of the ensemble of Hamiltonians may be constructed from [15,22]…”

“…In this Section, we present numerical simulations of the twopoint GUE correlation and cluster functions defined by (14) and (15) with the delta function represented by (9). Some details on how the ensemble average in (14) was coded are given in the Appendix.…”

“…(The region E x < 0 is not shown since T 2 (E x , E y ) = T (E y , E x )).) The numerical simulations of T 2 (E x , E y ) (solid black curves) were carried out using (15) and (9) for the same values of and k max as Fig. 2.…”

“…Random matrices were introduced by Wishart in multivariate statistics in the 1920s [1] and Wigner in the study of neutron resonances in the 1950s [2]. Since then random matrix theory has found further applications in nuclear physics [3][4][5][6][7][8] as well as in quantum and wave chaos [9,10], quantum chromodynamics [11], mesoscopic physics [12,13], quantum gravity [14], numerical computation [15], number theory [16][17][18] and complex systems [19].…”

Применение Теории Случайных Матриц Для Описания Бозонного Пика В Двумерных Системах

“…Поэтому целью нашей работы является применение теории случайных матриц для анализа колебательных свойств аморфных твердых тел. Теория случайных матриц имеет важные приложения во многих различных областях науки и техники при анализе сложных систем, состоящих из большого числа степеней свободы [14][15][16][17][18][19][20][21][22][23][24]. Однако исследуемая колебательная система имеет ряд отличительных особенностей.…”