Operators and special functions in random matrix theory (2008).Operators and special functions in random matrix theory (2008).

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 Author Msci, Andrew Mccafferty ♦ Allen, Woody Source CiteSeerX Content type Text File Format PDF
 Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science Subject Keyword Random Matrix Theory ♦ Integral Operator ♦ Special Function ♦ Fredholm Determinant ♦ Han-kel Operator ♦ Differential Operator ♦ Sufficient Condition ♦ Tracy Widom Operator ♦ Jacobi Kernel Converges ♦ Tracy Widom Type ♦ Toeplitz Operator ♦ Trace Norm ♦ Hankel Square ♦ New Example ♦ Hard Edge Scaling ♦ Operator Exists ♦ Differential Equation Coef-ficients ♦ Discrete Tw Operator ♦ Analogous Operator ♦ Bessel Kernel ♦ Discrete Random Matrix Ensemble ♦ Tw Integral Operator ♦ Unsolvable Equation ♦ Probabilistic Calculation Abstract The Fredholm determinants of integral operators with kernel of the form A(x)B(y) −A(y)B(x) x − y arise in probabilistic calculations in Random Matrix Theory. These were ex-tensively studied by Tracy and Widom, so we refer to them as Tracy–Widom operators. We prove that the integral operator with Jacobi kernel converges in trace norm to the integral operator with Bessel kernel under a hard edge scaling, using limits derived from convergence of differential equation coef-ficients. The eigenvectors of an operator with kernel of Tracy–Widom type can sometimes be deduced via a commuting differential operator. We show that no such operator exists for TW integral operators acting on L2(R). There are analogous operators for discrete random matrix ensembles, and we give sufficient conditions for these to be expressed as the square of a Han-kel operator: writing an operator in this way aids calculation of Fredholm determinants. We also give a new example of discrete TW operator which can be expressed as the sum of a Hankel square and a Toeplitz operator. Previously unsolvable equations are dealt with by threats of reprisals... Educational Role Student ♦ Teacher Age Range above 22 year Educational Use Research Education Level UG and PG ♦ Career/Technical Study Publisher Date 2008-01-01