Linear Differential Equations for the Resolvents of the Classical Matrix Ensembles

Anas A. Rahman; Peter J. Forrester

REVIEW 3 major objections 4 minor 72 references

Linear Differential Equations for the Resolvents of the Classical Matrix Ensembles

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The spectral density of every classical β-ensemble is an exact solution of a linear differential equation of order β+1, and the paper writes these equations down explicitly for β=1, 2, 4, 6, and 2/3.

desk verdict Genuinely useful paper with new explicit differential equations for classical β-ensemble densities and resolvents; the flagship β=4 and β=6 formulas rest on suppressed computer algebra that referees should ask to see. read the letter →

arxiv 1908.04963 v3 pith:NCDDHK2V submitted 2019-08-14 math-ph math.MP

Anas A. Rahman , Peter J. Forrester This is my paper · ORCID

classification math-phmath.MP MSC 15B5233C4534A3060B20

keywords betaensemblesspectraldensitylineardifferentialequationsresolventsSelbergcorrelationintegralsdualityrelationsmomentsrandommatrixtheory

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes that for the Gaussian, Laguerre, and Jacobi β-ensembles the spectral density is annihilated by a linear differential operator of order β+1, and that the resolvent satisfies the same operator acting on N⁻¹ times the resolvent, equal to an explicit inhomogeneous term. Explicit operators are given for β=2 and 4 in all three classical settings, for β=6 in the Gaussian case, and the β↔4/β duality supplies the companion β=1 and Gaussian β=2/3 results. These differential characterizations are then used to derive recurrences for spectral moments, first-order differential equations for the coefficients of the 1/N expansions of the resolvents, and edge-scaled ODEs. A sympathetic reader would care because this unifies previously scattered third- and fifth-order equations and extends them to new cases such as β=6 and β=2/3.

What carries the argument

The engine is the Selberg correlation integral $J^{{(N)}}$_{n,p}(x), an average of products of characteristic-polynomial factors, which satisfies the first-order differential-difference system (2.11). For even β, the spectral-density average $I^{{(J)}}$_{β,N}(x) is proportional to (-x)^{βN} $J^{{(N)}}$_{β,0}(1/x), so the components p=0,1,...,β form a closed matrix differential equation; eliminating the auxiliary components yields a scalar ODE of order β+1 for the density. The same structure, together with the duality (2.7) and analytic continuation to parameters with b′<−1, carries over to β=1, while the Gaussian and Laguerre cases are obtained by limiting procedures. The Stieltjes transform then turns the density ODE into the resolvent's inhomogeneous ODE.

What would settle it

Take a Jacobi parameter pair with b′<−1, compute $J^{{(N)}}$_{2,0}(x) by an independent method such as the closed form (2.8) in terms of the multivariate hypergeometric function, and check whether it satisfies the differential equation (2.14); equivalently, evaluate $ρ^{{(J)}}$_{(1),1,N}(x) by direct numerical quadrature for N=3 and test whether $D^{{(J)}}$_{1,N}ρ=0 holds.

Watch

Extended reading notes

Core claim

The central discovery is that for each classical weight, the averaged power of the characteristic polynomial, and hence the spectral density itself, solves a linear homogeneous ODE of order β+1. For example, in the Jacobi case with β=2, the operator $D^{{(J)}}$_{2,N} satisfies $D^{{(J)}}$_{2,N} $ρ^{{(J)}}$_{(1),2,N}=0, while applying the same operator to N⁻$¹W^{{(J)}}$_{2,N} gives an explicit polynomial right-hand side; analogous statements hold for β=4 and for the Laguerre and Gaussian ensembles. The paper obtains these by substituting the Selberg correlation integral representation into a first-order matrix differential system and eliminating auxiliary components, then uses dualities to reach β=1 and β=2/3. It also derives the corresponding moment recurrences, first-order differential equations for the 1/N expansion coefficients of the resolvents, and the edge-scaled equations.

Load-bearing premise

The load-bearing premise is that the differential-difference system (2.11) for Selberg correlation integrals remains valid under the analytic continuation that sends the Jacobi parameter b′ below −1; if this fails, the derived β=1 and β=4 Jacobi equations, and the Laguerre and Gaussian equations built from them, do not follow.

Editorial extensions

If this is right

Spectral moments of the Jacobi ensemble obey a third-order linear recurrence for β=2 and a fifth-order one for β=1,4, with analogous Laguerre and Gaussian recurrences; these recover known results and give new ones for β=6 and β=2/3.
The 1/N expansion coefficients W_l^β(x) of the scaled resolvents satisfy first-order differential-difference equations, providing a recursive generation scheme for these coefficients that avoids multi-point correlators.
At the soft edge, the scaled densities for β=2/3, 1, 2, 4, and 6 satisfy a single explicit differential equation D^{soft}_{β,∞}ρ^{soft}=0, and at the hard edge the β=1, 2, and 4 densities satisfy D^{hard}_{β,∞}ρ^{hard}=0.
The differential equations characterize the density even in cases where no closed-form expression is known, such as the β=2/3 soft-edge density.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Inference: The paper's elimination scheme is in principle available for any even β, so explicit seventh-order equations for the Laguerre and Jacobi ensembles at β=6 should be derivable by the same limiting procedure, not just for the Gaussian case.
Inference: The observed β-independence of the leading resolvent coefficient W_0^β(x) suggests that the higher coefficients may organize by powers of h=√κ−1/√κ; testing this pattern for the new β=6 and β=2/3 coefficients would sharpen the structural picture.
Inference: A direct numerical check of the Jacobi equations at parameters with b′<−1, beyond the N=1 and N=2 checks reported in the paper, would independently test whether the analytic-continuation step underlying the β=1 Jacobi result is sound.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper delivers more than the title promises: a uniform Selberg-integral/duality route to linear ODEs for the spectral densities and resolvents in all three classical ensembles, for β=1,2,4, and new seventh-order Gaussian equations for β=6 and 2/3. The β=2 and β=1,4 Gaussian cases were known, but the Jacobi and Laguerre β=1,4 equations and the β=6/2/3 Gaussian equations are new, and the moment recursions and topological-expansion coefficient equations are useful side products. The derivation from the differential-difference system (2.11) is shown in full for β=2, and the duality arguments for β=1 are elegant and consistent with existing moment data. The paper is honest about what it checks and what it does not.

The soft spot is exactly where the stress-test puts it. The explicit fifth-order Jacobi operator in Theorem 2.2 and the seventh-order Gaussian operator in Proposition 2.3 are the central deliverables, but their derivation from the matrix systems (2.18) and (2.41) is asserted, not shown. Lemma 2.2 states that the lower-derivative coefficients are suppressed and that computer algebra is the efficient route; Proposition 2.3 says 'then yields' with no displayed algebra. Since a single transcription error would falsify the stated formula, this is an omitted proof of the main result, not a style quibble. That said, the paper does provide consistency checks — N=1,2, large-N limits, agreement with topological expansion coefficients up to l=6 in Proposition 3.11 — and the method is systematic, so my prior is that the formulas are correct. But a referee should not have to take that on faith. The analytic-continuation worry about (2.11) for b′<−1 is minor: the identity is polynomial-coefficient and extends by analyticity once the Selberg integrals are defined. The reliance on Forrester's earlier differential-difference system is legitimate; it is not circular.

Who should read this: anyone working on β-ensemble asymptotics, moment recursions, or edge universality. The paper deserves a serious referee. My recommendation: send it out, and require the authors to supply the full coefficient algebra or the computer-algebra notebook for Lemma 2.2 and Proposition 2.3 before acceptance. With that, it's a solid contribution.

Referee Report

3 major / 4 minor

Summary. The paper derives linear differential equations of order β+1 satisfied by the spectral densities of the classical Gaussian, Laguerre, and Jacobi β-ensembles, together with companion inhomogeneous equations for the resolvents. The explicit equations are given for β=2 and β=4 in all three classical cases, and for β=6 in the Gaussian case, with the cases β=1 and β=2/3 obtained via known β↔4/β dualities. The method uses Selberg correlation integrals: the differential-difference system (2.11) is converted to a matrix differential system whose elimination yields the scalar equation. Applications include recurrences for spectral moments and their 1/N expansion coefficients, first-order differential equations for the resolvent expansion coefficients in the topological expansion, and differential equations characterizing the soft- and hard-edge scaled densities.

Significance. If the displayed formulas are correct, this is a substantial contribution: it unifies and extends known β=2 results (Götze–Tikhomirov, Ledoux, Haagerup–Thorbjørnsen) and gives new explicit characterizations for β=1, β=4, and, in the Gaussian case, β=6 and β=2/3. The framework is principled, resting on external prior results (the Forrester differential-difference system, Selberg integral dualities, and known moment formulas) rather than on the target differential equations themselves. The applications to moment recurrences and to 1/N expansion coefficients (Harer–Zagier type recursions) are of independent interest, and the paper honestly reports machine checks and consistency with topological expansions. However, a substantial part of the central derivation is suppressed computer algebra, and the paper does not make that algebra reproducible.

major comments (3)

[Lemma 2.2 and Theorem 2.2 (§2.1)] The derivation of the β=4 Jacobi differential equation is not actually presented. Lemma 2.2 explicitly states that the lower-order coefficient forms are 'suppressed' and are 'obtained most efficiently by computer algebra', and the proof of Theorem 2.2 says one 'undertakes the same steps' as in the β=2 proof without displaying the elimination from the 5×5 system (2.18). Since the content of Theorem 2.2 is exactly the full coefficient formula (2.23), this is an omitted verification of the paper's central claim. Please include the full elimination, or provide a computer algebra script/ancillary file that produces (2.23) and (2.26) from (2.18), along with the verification commands.
[Proposition 2.3 (§2.3)] The step from the 7×7 matrix differential system (2.41) to the seventh-order scalar operator (2.38) is asserted with the phrase 'then yields', and no intermediate algebra is displayed. As with the β=4 Jacobi case, this is a load-bearing omitted proof: the displayed operator (2.38) is the main deliverable of that subsection. Please supply the elimination details or a machine-readable derivation that the reader can execute to reproduce (2.38) and (2.40).
[Verification statements throughout §2 and §3.3] The statements that equations have been 'checked for N=1,2 using computer algebra' and that Proposition 3.11 'has been checked against [72] up to l=6' are not reproducible from the manuscript because no code or detailed outputs are provided. Given the length of (2.23), (2.38), and the recurrence coefficients in §3, a transcription error is a real possibility and the verification is essential. Please provide the computer algebra code, or at minimum the explicit commands and a small sample of output, so the checks can be independently repeated.

minor comments (4)

[Equation (2.38) and (2.40)] The operator for β=2/3 contains powers such as (κ−1)^{7/2} with κ=1/3, i.e. negative base raised to half-integer powers; the branch of (κ−1)^{1/2} and the sense in which the resulting operator is real on the space of even densities should be specified.
[Equation (3.9)] The recurrence in Proposition 3.4 is written as a bare sum; the right-hand side '= 0' is missing. Also, the indices in the display and the stated ranges 'k>12' and 'k>6' in Proposition 3.5 should be checked for consistency.
[Notation in §3.3] The notation for the zero extension of the expansion coefficients is inconsistent, e.g. 'W (G),k β := 0' appears with italic and roman subscripts in the same sentence; this should be cleaned up.
[After (2.7), analytic continuation] The text notes that b' may be less than −1 and that the Selberg integral must be interpreted by analytic continuation; since (2.11) is a differential identity with polynomial coefficients in the parameters, a one-sentence justification of why the identity extends to this continued regime would make the derivation fully airtight.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the differential equations follow from external Selberg-correlation identities, known dualities, and published moment formulas; suppressed computer algebra is a verification gap, not circularity.

full rationale

The paper's central derivations are not circular. The homogeneous density equations (2.20), (2.25), (2.30), and (2.39) are obtained by applying the external differential-difference system (2.11), cited to Forrester [28] and also to the book [31], to the Selberg correlation integrals J_{n,p}^{(N)}. That system is a parameter-free identity whose assumptions do not include the target resolvent or density equations, so it is independent support rather than a self-citation chain. The inhomogeneous resolvent equations (2.21), (2.26), (2.31), and (2.40) are derived by Stieltjes-transforming the homogeneous density equations and substituting known spectral moments from [24, 61, 72]; these are external benchmarks, not fitted parameters renamed as predictions. The beta=1 and beta=2/3 cases are imported through independent duality relations [23, 25, 38, 43], not through equations whose content already contains the desired conclusion. What the manuscript does leave unverified from the text alone is the scalar elimination for beta=4 and beta=6: Lemma 2.2 explicitly suppresses the lower-order coefficients of (2.17) and refers to computer algebra, and Proposition 2.3 states that substitution in (2.41) 'yields' the seventh-order operator (2.38) without displaying the elimination. That is an omitted-proof and reproducibility concern, not a circularity: the asserted operators are not defined in terms of the quantities they are supposed to predict, and no fitted data are being recycled as a prediction. The analytic-continuation issue noted after (2.8) is also not circular, since (2.11) is a polynomial-coefficient identity that extends by analytic continuation once the Selberg integrals are defined. Overall, the derivation chain is self-contained against external inputs, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper's central claim rests on prior results about Selberg correlation integrals and β↔4/β dualities, all cited from the literature, plus a stated analytic continuation and a uniqueness assertion. No parameters are fitted to data; the moments used to fix inhomogeneous terms are external inputs.

assumptions (5)

domain assumption The differential-difference system (2.11) for the Selberg correlation integrals J(N)n,p(x), as given in Forrester [28].
All scalar differential equations in Section 2 are obtained by eliminating the auxiliary functions J(N)n,p from this system; the paper cites [28] for (2.11) and does not prove it.
domain assumption The duality formulas for characteristic-polynomial averages: Gaussian (1.5) from [3] and Jacobi (2.7) from [31, Ch. 13], and the β↔4/β moment dualities used for β=1 and 2/3.
Used to carry β=2,4,6 results to β=1 and 2/3; quoted from prior literature (Baker-Forrester, Forrester, Dumitriu-Edelman, Fyodorov-Le Doussal).
domain assumption The identity ρ(J)(1),β,N+1(x) ∝ x−a−βN(1−x)−b J(N)β,0(1/x) from (1.3), (2.7), (2.10).
Bridges the Selberg average to the spectral density; derived from Selberg integral evaluations, itself a prior result.
domain assumption Analytic continuation of the Jacobi average to parameters b′ < −1, as stated after (2.8).
The Jacobi duality (2.7) introduces b′ = −(b+n)/κ − N which can be < −1; the average on the right is taken by analytic continuation, and (2.11) is assumed to hold in this regime.
domain assumption Uniqueness of the polynomial solution consistent with the leading large-x behavior (2.5).
Remark 2.1(3) asserts the ODE admits a unique polynomial solution determined by c0; this uniqueness is stated without proof and is needed for the characterization.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Linear Differential Equations for the Resolvents of the Classical Matrix Ensembles." pith.science (2026). https://pith.science/paper/NCDDHK2V

@misc{pith2026190804963,
  author       = {Pith},
  title        = {Pith review of: Linear Differential Equations for the Resolvents of the Classical Matrix Ensembles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NCDDHK2V}},
  note         = {Machine review of arXiv:1908.04963}
}

abstract

The spectral density for random matrix $\beta$ ensembles can be written in terms of the average of the absolute value of the characteristic polynomial raised to the power of $\beta$, which for even $\beta$ is a polynomial of degree $\beta(N-1)$. In the cases of the classical Gaussian, Laguerre, and Jacobi weights, we show that this polynomial, and moreover the spectral density itself, can be characterised as the solution of a linear differential equation of degree $\beta+1$. This equation, and its companion for the resolvent, are given explicitly for $\beta=2$ and $4$ for all three classical cases, and also for $\beta=6$ in the Gaussian case. Known dualities for the spectral moments relating $\beta$ to $4/\beta$ then imply corresponding differential equations in the case $\beta=1$, and for the Gaussian ensemble, the case $\beta=2/3$. We apply the differential equations to give a systematic derivation of recurrences satisfied by the spectral moments and by the coefficients of their $1/N$ expansions, along with first-order differential equations for the coefficients of the $1/N$ expansions of the corresponding resolvents. We also present the form of the differential equations when scaled at the hard or soft edges.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

72 extracted references · 72 canonical work pages

[72]

Witte and P .J

N.S. Witte and P .J. Forrester,Moments of the Gaussian β ensembles and the large N expansion of the densities, J. Math. Phys. 55 (2014), 083302. 3, 4, 5, 6, 18, 22, 23, 28, 29, 30 School of Mathematics and Statistics, ARC Centre of Excellence for Mathematical and Statisti- cal Frontiers, University of Melbourne, Victoria 3010, Australia E-mail address: an...

work page 2014
[1]

Adachi, M

S. Adachi, M. Toda, and H. Kubotani, Asymptotic analysis of singular values of rectangular complex matrices in the Laguerre and ﬁxed trace ensembles , J. Phys. A 44 (2011), 292002(8pp). 3, 4

work page 2011
[2]

Adler, P .J

M. Adler, P .J. Forrester, T. Nagao, and P . van Moerbeke, Classical skew orthogonal polynomials and random matrices, J. Stat. Phys. 99 (2000), 141–170. 4, 23

work page 2000
[3]

Baker and P .J

T.H. Baker and P .J. Forrester, The Calogero-Sutherland model and generalized classical polynomials , Commun. Math. Phys. 188 (1997), 175–216. 2 40 ANAS A. RAHMAN AND PETER J. FORRESTER

work page 1997
[4]

Baker, P .J

T.H. Baker, P .J. Forrester, and P .A. Pearce,Random matrix ensembles with an extensive external charge , J. Phys. A 31 (1998), 6087. 34

work page 1998
[5]

Beenakker, Random-matrix theory of quantum transport, Rev

C.W.J. Beenakker, Random-matrix theory of quantum transport, Rev. Mod. Phys. 69 (1997), 731–808. 26

work page 1997
[6]

Borot and A

G. Borot and A. Guionnet, Asymptotic expansion of β matrix models in the one-cut regime, Commun. Math. Phys. 317(2) (2013), 447–483. 5

work page 2013
[7]

Brack, A

M. Brack, A. Koch, M.V .N. Murthy and J. Roccia, Exact and asymptotic local virial theorems for ﬁnite fermionic systems, J. Phys. A: Math. Theor. 43 (2010), 255204. 34

work page 2010

Show all 72 references

[8]

Brack and M.V .N

M. Brack and M.V .N. Murthy, Harmonically trapped fermion gases: exact and asymptotic results in arbitrary dimension, J. Phys. A 36 (2003), 1111–1133. 3

2003
[9]

Brouwer, K.M

P .W. Brouwer, K.M. Frahm, and C.W.J. Beenakker,Quantum mechanical time-delay matrix in chaotic scattering , Phys. Rev. Lett. 78 (1997), 4737–4740. 19, 26

1997
[10]

Bryc and V

W. Bryc and V . Pierce,Duality of real and quaternionic random matrices , Elec. J. Probab. 14 (2009), 452–476. 19

2009
[11]

Chaudhuri and N

A. Chaudhuri and N. Do, Generalisations of the Harer-Zagier recursion for 1-point functions, arXiv:1812.11885. 24

arXiv
[12]

Cohen, F.D

P . Cohen, F.D. Cunden and N. O’Connell, Moments of discrete orthogonal polynomial ensembles , arXiv:1907.12884. 4

1907 arXiv
[13]

Cunden, A

F.D. Cunden, A. Dahlqvist and N. O’Connell, Integer moments of complex Wishart matrices and Hurwitz numbers, arXiv:1809.10033. 4

arXiv
[14]

Cunden, F

F.D. Cunden, F. Mezzadri, N. O’Connell and N. Simm, Moments of random matrices and hypergeometric orthogonal polynomials, Commun. Math. Phys. ( 2019), https://doi.org/10.1007/s00220-019-03323-9 4, 11, 19, 20, 23

2019 doi
[15]

Cunden, F

F.D. Cunden, F. Mezzadri, N. Simm, and P . Vivo,Correlators for the Wigner-Smith time-delay matrix of chaotic cavities, J. Phys. A 49 (2016), 18LT01. 4, 26

2016
[16]

Cunden, F

F.D. Cunden, F. Mezzadri, N. Simm, and P . Vivo,Large-N expansion for the time-delay matrix of ballistic chaotic cavities, J. Math. Phys. 57 (2016), 111901. 4, 5, 19, 21, 22, 23, 26

2016
[17]

D.S. Dean, P . Le Doussal, S.N. Majumdar and G. Schehr, Non-interacting fermions at ﬁnite temperature in a d-dimensional trap: universal correlations , Phys. Rev. A 94 (2016), 063622. 34

2016
[18]

Desrosiers, Duality in random matrix ensembles for all β, Nucl

P . Desrosiers, Duality in random matrix ensembles for all β, Nucl. Phys. B 817 (2009), 224–251. 2

2009
[19]

Desrosiers and P .J

P . Desrosiers and P .J. Forrester,Hermite and Laguerre β-ensembles: asymptotic corrections to the eigenvalue density, Nucl. Phys. B 743 (2006), 307–332. 35

2006
[20]

Desrosiers and D.-Z

P . Desrosiers and D.-Z. Liu, Selberg integrals, super-hypergeometric functions and applications to β-ensembles, Random Matrices: Theory Appl. 4 (2015), 1550007. 2

2015
[21]

Di Francesco, Rectangular matrix models and combinatorics of colored graphs , Nucl

P . Di Francesco, Rectangular matrix models and combinatorics of colored graphs , Nucl. Phys. B 648 (2003), 461–496. 5, 19, 25

2003
[22]

Dumaz and B

L. Dumaz and B. Virág, The right tail exponent of the Tracy-Widom β distribution, Ann. Inst. Henri Poincaré Probab. Stat. 49 (2013), 915–933. 35

2013
[23]

Dumitriu and A

I. Dumitriu and A. Edelman, Global spectrum ﬂuctuations for the β-Hermite and β-Laguerre ensembles via matrix models, J. Math. Phys. 47 (2006), 063302. 3, 5, 6, 14, 15, 18, 24

2006
[24]

Dumitriu, A

I. Dumitriu, A. Edelman, and G. Shuman, MOPS: Multivariate orthogonal polynomials (symbolically) , J. Symb. Comput. 42 (2007), 587–620. 13, 22

2007
[25]

Dumitriu and E

I. Dumitriu and E. Paquette, Global ﬂuctuations for linear statistics of β Jacobi ensembles, Random Matrices: Theory Appl. 01 (2012), 1250013. 3, 5, 13, 24 LINEAR DE’S FOR SOME RANDOM MATRIX RESOLVENTS 41

2012
[26]

Eynard and N

B. Eynard and N. Orantin, Topological recursion in enumerative geometry and random matrices, J. Phys. A 42 (2009), 293001. 5, 19

2009
[27]

Forrester,Selberg correlation integrals and the 1/r2 quantum many body system, Nucl

P .J. Forrester,Selberg correlation integrals and the 1/r2 quantum many body system, Nucl. Phys. B 388 (1992), 671–699. 2, 3

1992
[28]

Forrester,Recurrence equations for the computation of correlations in the 1/r2 quantum many body system, J

P .J. Forrester,Recurrence equations for the computation of correlations in the 1/r2 quantum many body system, J. Stat. Phys. 72 (1993), 39–50. 8

1993
[29]

Forrester,Exact results and universal asymptotics in the Laguerre random matrix ensemble , J

P .J. Forrester,Exact results and universal asymptotics in the Laguerre random matrix ensemble , J. Math. Phys. 35 (1993), 2539–2551. 2, 3, 35

1993
[30]

Forrester, The spectrum edge of random matrix ensembles , Nucl

P .J. Forrester, The spectrum edge of random matrix ensembles , Nucl. Phys. B 402 (1993), 709–728. 34

1993
[31]

Forrester, Log-gases and random matrices, Princeton University Press, Princeton, NJ, ( 2010)

P .J. Forrester, Log-gases and random matrices, Princeton University Press, Princeton, NJ, ( 2010). 2, 3, 7, 8, 11

2010
[32]

Forrester,Large deviation eigenvalue density for the soft edge Laguerre and Jacobi β-ensembles, J

P .J. Forrester,Large deviation eigenvalue density for the soft edge Laguerre and Jacobi β-ensembles, J. Phys. A 45 (2012), 145201. 33, 34

2012
[33]

Forrester,Spectral density asymptotics for Gaussian and Laguerre β-ensembles in the exponentially small region, J

P .J. Forrester,Spectral density asymptotics for Gaussian and Laguerre β-ensembles in the exponentially small region, J. Phys. A 45 (2012), 075206. 35

2012
[34]

Forrester, Moments of the ground state density for the d-dimensional Fermi gas in an harmonic trap , arXiv:1909.09918 3

P .J. Forrester, Moments of the ground state density for the d-dimensional Fermi gas in an harmonic trap , arXiv:1909.09918 3

1909 arXiv
[35]

Forrester, N.E

P .J. Forrester, N.E. Frankel, and T.M. Garoni,Asymptotic form of the density proﬁle for Gaussian and Laguerre random matrix ensembles with orthogonal and symplectic symmetry , J. Math. Phys. 47 (2006), 023301. 35

2006
[36]

Forrester and A

P .J. Forrester and A. Gamburd, Counting formulas associated with some random matrix averages , J. Combin. Theory Ser. A 113 (2006), 934-951. 1

2006
[37]

Forrester and M

P .J. Forrester and M. Ito, Difference system for Selberg correlation integrals, J. Phys. A 43 (2010), 175202. 2

2010
[38]

Forrester, A.A

P .J. Forrester, A.A. Rahman, and N.S. Witte, Large N expansions for the Laguerre and Jacobi β ensembles from the loop equations, J. Math. Phys. 58 (2017), 113303. 3, 5, 13, 14, 15, 19, 20, 22, 28, 29

2017
[39]

Forrester and E.M

P .J. Forrester and E.M. Rains, A Fuchsian matrix differential equation for Selberg correlation integrals , Commun. Math. Phys. 309 (2012), 771–792. 2, 8

2012
[40]

Forrester and A.K

P .J. Forrester and A.K. Trinh,Functional form for the leading correction to the distribution of the largest eigenvalue in the GUE and LUE, J. Math. Phys. 59 (2018), 053302. 11, 34, 35, 36

2018
[41]

Forrester and A.K

P .J. Forrester and A.K. Trinh,Optimal soft edge scaling variables for the Gaussian and Laguerre even β ensembles, Nucl. Phys. B 938 (2019), 621-639. 16, 36

2019
[42]

Forrester and A.K

P .J. Forrester and A.K. Trinh, Finite Size Corrections at the Hard Edge for the Laguerre β ensemble, arXiv:1903.08823. 37

1903 arXiv
[43]

Fyodorov and P

Y.V . Fyodorov and P . Le Doussal,Moments of the position of the maximum for GUE characteristic polynomials and for log-correlated Gaussian processes, J. Stat. Phys. 164 (2016), 190–240. 3, 13, 14

2016
[44]

Garoni, P .J

T.M. Garoni, P .J. Forrester, and N.E. Frankel, Asymptotic corrections to the eigenvalue density of the GUE and LUE, J. Math. Phys. 46 (2005), 103301. 29

2005
[45]

Gaberdiel, A

M.R. Gaberdiel, A. Klemm, and I. Runkel, Matrix model eigenvalue integrals and twist ﬁelds in the su(2)-WZW model, J. High Energy Phys. 2005 (2005), JHEP10(2005). 11

2005
[46]

Götze and A

F. Götze and A. Tikhomirov, The rate of convergence for spectra of GUE and LUE matrix ensembles , Cent. Eur. J. Math. 3 (2005), 666–704. 3, 4

2005
[47]

Haagerup and S

U. Haagerup and S. Thorbjørnsen, Random matrices with complex Gaussian entries , Expo. Math. 21 (2003), 293–337. 3, 4 42 ANAS A. RAHMAN AND PETER J. FORRESTER

2003
[48]

Haagerup and S

U. Haagerup and S. Thorbjørnsen, Asymptotic Expansions for the Gaussian Unitary Ensemble , Inﬁnite Dim. Anal., Quantum Probab. and Related Topics 15 (2012), 1250003. 29, 33

2012
[49]

Harer and D

J. Harer and D. Zagier, The Euler characteristic of the moduli space of curves , Inven. Math. 85 (1986), 457–485. 4, 5, 19, 22, 24

1986
[50]

Holcomb and G.R.M

D. Holcomb and G.R.M. Flores, Edge Scaling of the β-Jacobi ensemble, J. Stat. Phys. 149 (2012), 1136–1160. 35

2012
[51]

Johnstone, On the distribution of the largest principal component , Ann

I.M. Johnstone, On the distribution of the largest principal component , Ann. Stat. 29 (2001), 295–327. 34

2001
[52]

Johnstone, Multivariate Analysis and Jacobi Ensembles: Largest Eigenvalue, Tracy-Widom limits and rates of convergence, Ann

I.M. Johnstone, Multivariate Analysis and Jacobi Ensembles: Largest Eigenvalue, Tracy-Widom limits and rates of convergence, Ann. Stat. 36 (2008), 2638–2716. 35

2008
[53]

Kaneko, Selberg integrals and hypergeometric functions associated with Jack polynomials , SIAM J

J. Kaneko, Selberg integrals and hypergeometric functions associated with Jack polynomials , SIAM J. Math Anal. 24 (1993), 1086–1110. 8

1993
[54]

Katori and N

M. Katori and N. Komatsuda, Moments of vicious walkers and Möbius graph expansions , Phys. Rev. E 67 (2003), 051110. 5, 24

2003
[55]

Kopelevitch, A convergent 1 N expansion for the GUE, Ann

O. Kopelevitch, A convergent 1 N expansion for the GUE, Ann. Henri Poincaré 19 (2018), 3883–3899. 3, 4

2018
[56]

LaCroix, The combinatorics of the Jack parameter and the genus series for topological maps , Ph.D

M.A. LaCroix, The combinatorics of the Jack parameter and the genus series for topological maps , Ph.D. thesis, The University of Waterloo (2009). 5, 19

2009
[57]

Lawes and N.H

G.P . Lawes and N.H. March,Exact local density method for linear harmonic oscillator , J. Chem. Phys. 71 (1979), 1007–1009. 3

1979
[58]

Ledoux, Differential operators and spectral distributions of invariant ensembles from the classical orthogonal polynomials

M. Ledoux, Differential operators and spectral distributions of invariant ensembles from the classical orthogonal polynomials. The continuous case , Electron. J. Probab. 9 (2004), 177–208. 3, 4, 5, 20, 21

2004
[59]

Ledoux, A recursion formula for the moments of the Gaussian orthogonal ensemble , Ann

M. Ledoux, A recursion formula for the moments of the Gaussian orthogonal ensemble , Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009), 754–769. 4, 5, 22, 23, 24

2009
[60]

Livan and P

G. Livan and P . Vivo,Moments of Wishart-Laguerre and Jacobi ensembles of random matrices: application to the quantum transport problem in chaotic cavities, Acta Phys. Pol. B 42 (2011), 1081–1104. 4, 26

2011
[61]

Mezzadri, A.K

F. Mezzadri, A.K. Reynolds and B. Winn, Moments of the eigenvalue densities and of the secular coefﬁcients of β-ensembles, Nonlinearity 30 (2015), 1034. 5, 11, 13, 20, 22, 24

2015
[62]

Mezzadri and N.J

F. Mezzadri and N.J. Simm, Moments of the transmission eigenvalues, proper delay times and random matrix theory I, J. Math. Phys. 52 (2011), 103511. 4, 19, 25, 26

2011
[63]

Mezzadri and N.J

F. Mezzadri and N.J. Simm, Moments of the transmission eigenvalues, proper delay times and random matrix theory II, J. Math. Phys. 53 (2012), 053504. 4, 19, 26

2012
[64]

Mulase and A

M. Mulase and A. Waldron, Duality of orthogonal and symplectic matrix integrals and quaternionic Feynman graphs, Commun. Math. Phys. 240 (2003), 553–586. 19

2003
[65]

Nagao and P .J

T. Nagao and P .J. Forrester,Asymptotic correlations at the spectrum edges of random matrices , Nucl. Phys. B 435 (1995), 401–420. 37

1995
[66]

Naprienko, A convergent 1/n-expansion for GSE and GOE, arXiv:1801.02359

Y. Naprienko, A convergent 1/n-expansion for GSE and GOE, arXiv:1801.02359. 4, 6

arXiv
[67]

Norbury and N

P . Norbury and N. Do,Topological recursion for irregular spectral curves, Journal of the London Math. Soc. 97(3) (2018), 398–426. 5, 26

2018
[68]

Novaes, Statistics of quantum transport in chaotic cavities with broken time reversal symmetry , Phys

M. Novaes, Statistics of quantum transport in chaotic cavities with broken time reversal symmetry , Phys. Rev. B 78 (2008), 035337. 4, 26

2008
[69]

Pastur and M

L. Pastur and M. Shcherbina, Eigenvalue distribution of large random matrices, American Mathematical Society, Providence, RI, (2011). 1, 25

2011
[70]

Vivo and E

P . Vivo and E. Vivo,Transmission eigenvalue densities and moments in chaotic cavities from random matrix theory , J. Phys. A 41 (2008), 122004. 4, 26 LINEAR DE’S FOR SOME RANDOM MATRIX RESOLVENTS 43

2008
[71]

Witte and P .J

N.S. Witte and P .J. Forrester,Asymptotic forms for hard and soft edge general β conditional gap probabilities, Nucl. Phys. B 859 (2012), 321–340. 33

2012

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [72]

Witte and P .J

N.S. Witte and P .J. Forrester,Moments of the Gaussian β ensembles and the large N expansion of the densities, J. Math. Phys. 55 (2014), 083302. 3, 4, 5, 6, 18, 22, 23, 28, 29, 30 School of Mathematics and Statistics, ARC Centre of Excellence for Mathematical and Statisti- cal Frontiers, University of Melbourne, Victoria 3010, Australia E-mail address: an...

work page 2014

[2] [1]

Adachi, M

S. Adachi, M. Toda, and H. Kubotani, Asymptotic analysis of singular values of rectangular complex matrices in the Laguerre and ﬁxed trace ensembles , J. Phys. A 44 (2011), 292002(8pp). 3, 4

work page 2011

[3] [2]

Adler, P .J

M. Adler, P .J. Forrester, T. Nagao, and P . van Moerbeke, Classical skew orthogonal polynomials and random matrices, J. Stat. Phys. 99 (2000), 141–170. 4, 23

work page 2000

[4] [3]

Baker and P .J

T.H. Baker and P .J. Forrester, The Calogero-Sutherland model and generalized classical polynomials , Commun. Math. Phys. 188 (1997), 175–216. 2 40 ANAS A. RAHMAN AND PETER J. FORRESTER

work page 1997

[5] [4]

Baker, P .J

T.H. Baker, P .J. Forrester, and P .A. Pearce,Random matrix ensembles with an extensive external charge , J. Phys. A 31 (1998), 6087. 34

work page 1998

[6] [5]

Beenakker, Random-matrix theory of quantum transport, Rev

C.W.J. Beenakker, Random-matrix theory of quantum transport, Rev. Mod. Phys. 69 (1997), 731–808. 26

work page 1997

[7] [6]

Borot and A

G. Borot and A. Guionnet, Asymptotic expansion of β matrix models in the one-cut regime, Commun. Math. Phys. 317(2) (2013), 447–483. 5

work page 2013

[8] [7]

Brack, A

M. Brack, A. Koch, M.V .N. Murthy and J. Roccia, Exact and asymptotic local virial theorems for ﬁnite fermionic systems, J. Phys. A: Math. Theor. 43 (2010), 255204. 34

work page 2010