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Gap Probabilities for Edge Intervals in Finite Gaussian and Jacobi Unitary Matrix Ensembles
Mathematical Physics · Submitted 23 Aug 2000
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The probabilities for gaps in the eigenvalue spectrum of the finite dimension $ N \times N $ random matrix Hermite and Jacobi unitary ensembles on some single and disconnected double intervals are found.
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The probabilities for gaps in the eigenvalue spectrum of the finite dimension $ N \times N $ random matrix Hermite and Jacobi unitary ensembles on some single and disconnected double intervals are found. These are cases where a reflection symmetry exists and the probability factors into two other related probabilities, defined on single intervals. Our investigation uses the system of partial differential equations arising from the Fredholm determinant expression for the gap probability and the differential-recurrence equations satisfied by Hermite and Jacobi orthogonal polynomials. In our study we find second and third order nonlinear ordinary differential equations defining the probabilities in the general $N$ case. For N=1 and N=2 the probabilities and thus the solution of the equations are given explicitly. An asymptotic expansion for large gap size is obtained from the equation in the Hermite case, and also studied is the scaling at the edge of the Hermite spectrum as $ N \to \infty $, and the Jacobi to Hermite limit; these last two studies make correspondence to other cases reported here or known previously. Moreover, the differential equation arising in the Hermite ensemble is solved in terms of an explicit rational function of a {Painlev\'e-V} transcendent and its derivative, and an analogous solution is provided in the two Jacobi cases but this time involving a {Painlev\'e-VI} transcendent.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The probabilities for gaps in the eigenvalue spectrum of the finite dimension $ N \times N $ random matrix Hermite and Jacobi unitary ensembles on some single and disconnected double intervals are found. | result 0.56 | 0.94 | 0.69 | asserted 1.00 |
| s1 | These are cases where a reflection symmetry exists and the probability factors into two other related probabilities, defined on single intervals. | result 0.73 | 0.49 | 0.09 | asserted 1.00 |
| s2 | Our investigation uses the system of partial differential equations arising from the Fredholm determinant expression for the gap probability and the differential-recurrence equations satisfied by Hermite and Jacobi orthogonal polynomials. | method 1.00 | 0.72 | 0.05 | asserted 0.99 |
| s3 | In our study we find second and third order nonlinear ordinary differential equations defining the probabilities in the general $N$ case. | result 0.92 | 0.96 | 0.27 | asserted 1.00 |
| s4 | For N=1 and N=2 the probabilities and thus the solution of the equations are given explicitly. | result 0.99 | 0.94 | 0.09 | asserted 1.00 |
| s5 | An asymptotic expansion for large gap size is obtained from the equation in the Hermite case, and also studied is the scaling at the edge of the Hermite spectrum as $ N \to \infty $, and the Jacobi to Hermite limit; these last two studies make correspondence to other cases reported here or known previously. | result 0.84 | 0.93 | 0.10 | asserted 1.00 |
| s6 | Moreover, the differential equation arising in the Hermite ensemble is solved in terms of an explicit rational function of a {Painlev\'e-V} transcendent and its derivative, and an analogous solution is provided in the two Jacobi cases but this time involving a {Painlev\'e-VI} transcendent. | result 0.95 | 0.94 | 0.25 | asserted 1.00 |
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Original abstract
The probabilities for gaps in the eigenvalue spectrum of the finite dimension $ N \times N $ random matrix Hermite and Jacobi unitary ensembles on some single and disconnected double intervals are found. These are cases where a reflection symmetry exists and the probability factors into two other related probabilities, defined on single intervals. Our investigation uses the system of partial differential equations arising from the Fredholm determinant expression for the gap probability and the differential-recurrence equations satisfied by Hermite and Jacobi orthogonal polynomials. In our study we find second and third order nonlinear ordinary differential equations defining the probabilities in the general $N$ case. For N=1 and N=2 the probabilities and thus the solution of the equations are given explicitly. An asymptotic expansion for large gap size is obtained from the equation in the Hermite case, and also studied is the scaling at the edge of the Hermite spectrum as $ N \to \infty $, and the Jacobi to Hermite limit; these last two studies make correspondence to other cases reported here or known previously. Moreover, the differential equation arising in the Hermite ensemble is solved in terms of an explicit rational function of a {Painlev\'e-V} transcendent and its derivative, and an analogous solution is provided in the two Jacobi cases but this time involving a {Painlev\'e-VI} transcendent.
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The factorization of the low temperature correlation functions $C(M,N)$ for $ν = -k$ with $M+N$ odd, $M ≤ N$ considered in this paper, corresponds to a factorization of Toeplitz determinants that has been seen in many papers, in particular miscellaneous contexts (random matrices, see [ref] ).
Factorization of Ising correlations C(M,N) for $ \nu= \, -k$ and M+N odd, $M \le N$, $T < T_c$ and their lambda extensions · Conclusion · 0.82
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