% --------------------------------------------------------------------------- % Extreme gaps: integrable ensembles % --------------------------------------------------------------------------- @article{BenArousBourgade2013, author = {Ben Arous, G{\'e}rard and Bourgade, Paul}, title = {Extreme gaps between eigenvalues of random matrices}, journal = {Ann. Probab.}, volume = {41}, number = {4}, pages = {2648--2681}, year = {2013}, doi = {10.1214/11-AOP710}, eprint = {1010.1294}, archivePrefix = {arXiv}, primaryClass = {math.PR} } @article{FengWei2025, author = {Feng, Renjie and Wei, Dongyi}, title = {Large gaps of {CUE} and {GUE}}, journal = {Ann. Probab.}, volume = {53}, number = {5}, pages = {1668--1702}, year = {2025}, doi = {10.1214/24-AOP1747}, eprint = {1807.02149}, archivePrefix = {arXiv}, primaryClass = {math.PR} } @article{FengWei2021, author = {Feng, Renjie and Wei, Dongyi}, title = {Small gaps of circular {$\beta$}-ensemble}, journal = {Ann. Probab.}, volume = {49}, number = {2}, pages = {997--1032}, year = {2021}, doi = {10.1214/20-AOP1468}, eprint = {1806.01555}, archivePrefix = {arXiv}, primaryClass = {math.PR} } @article{FengTianWei2019, author = {Feng, Renjie and Tian, Gang and Wei, Dongyi}, title = {Small gaps of {GOE}}, journal = {Geom. Funct. Anal.}, volume = {29}, number = {6}, pages = {1794--1827}, year = {2019}, doi = {10.1007/s00039-019-00520-5}, eprint = {1901.01567}, archivePrefix = {arXiv}, primaryClass = {math.PR} } @misc{FengLiYao2024, author = {Feng, Renjie and Li, Jiaming and Yao, Dong}, title = {Small gaps of {GSE}}, howpublished = {Preprint, arXiv:2409.03324}, year = {2024}, eprint = {2409.03324}, archivePrefix = {arXiv}, primaryClass = {math.PR} } @misc{Charlier2026, author = {Charlier, Christophe}, title = {Largest gaps between bulk eigenvalues of unitary-invariant random {H}ermitian matrices}, howpublished = {Preprint, arXiv:2602.07524}, year = {2026}, eprint = {2602.07524}, archivePrefix = {arXiv}, primaryClass = {math.PR} } @phdthesis{Vinson2001, author = {Vinson, Jade P.}, title = {Closest spacing of eigenvalues}, school = {Princeton University}, year = {2001}, note = {Posted 2011 as arXiv:1111.2743}, eprint = {1111.2743}, archivePrefix = {arXiv}, primaryClass = {math.PR} } @article{Soshnikov2005, author = {Soshnikov, Alexander}, title = {Statistics of extreme spacing in determinantal random point processes}, journal = {Mosc. Math. J.}, volume = {5}, number = {3}, pages = {705--719}, year = {2005}, eprint = {math/0506286}, archivePrefix = {arXiv}, primaryClass = {math.PR} } @article{ValkoVirag2010, author = {Valk{\'o}, Benedek and Vir{\'a}g, B{\'a}lint}, title = {Large gaps between random eigenvalues}, journal = {Ann. Probab.}, volume = {38}, number = {3}, pages = {1263--1279}, year = {2010}, doi = {10.1214/09-AOP508}, eprint = {0811.0007}, archivePrefix = {arXiv}, primaryClass = {math.PR} } @article{ValkoVirag2009, author = {Valk{\'o}, Benedek and Vir{\'a}g, B{\'a}lint}, title = {Continuum limits of random matrices and the {B}rownian carousel}, journal = {Invent. Math.}, volume = {177}, number = {3}, pages = {463--508}, year = {2009}, doi = {10.1007/s00222-009-0180-z}, eprint = {0712.2000}, archivePrefix = {arXiv}, primaryClass = {math.PR} } % --------------------------------------------------------------------------- % Universality / comparison for extreme gaps % --------------------------------------------------------------------------- @article{Bourgade2022, author = {Bourgade, Paul}, title = {Extreme gaps between eigenvalues of {W}igner matrices}, journal = {J. Eur. Math. Soc. (JEMS)}, volume = {24}, number = {8}, pages = {2823--2873}, year = {2022}, doi = {10.4171/JEMS/1141}, eprint = {1812.10376}, archivePrefix = {arXiv}, primaryClass = {math.PR} } @article{LandonLopattoMarcinek2020, author = {Landon, Benjamin and Lopatto, Patrick and Marcinek, Jake}, title = {Comparison theorem for some extremal eigenvalue statistics}, journal = {Ann. Probab.}, volume = {48}, number = {6}, pages = {2894--2919}, year = {2020}, doi = {10.1214/20-AOP1439}, eprint = {1812.10022}, archivePrefix = {arXiv}, primaryClass = {math.PR} } @article{FigalliGuionnet2016, author = {Figalli, Alessio and Guionnet, Alice}, title = {Universality in several-matrix models via approximate transport maps}, journal = {Acta Math.}, volume = {217}, number = {1}, pages = {81--176}, year = {2016}, doi = {10.1007/s11511-016-0142-4}, eprint = {1407.2759}, archivePrefix = {arXiv}, primaryClass = {math.PR} } @article{NguyenTaoVu2017, author = {Nguyen, Hoi and Tao, Terence and Vu, Van}, title = {Random matrices: tail bounds for gaps between eigenvalues}, journal = {Probab. Theory Related Fields}, volume = {167}, number = {3-4}, pages = {777--816}, year = {2017}, doi = {10.1007/s00440-016-0693-5}, eprint = {1504.00396}, archivePrefix = {arXiv}, primaryClass = {math.PR} } @article{Zhang2025, author = {Zhang, Albert}, title = {Quantitative gap universality for {W}igner matrices}, journal = {Electron. J. Probab.}, volume = {31}, pages = {Paper No. 150}, year = {2026}, doi = {10.1214/26-EJP1601}, eprint = {2507.20442}, archivePrefix = {arXiv}, primaryClass = {math.PR} } @article{ErdosYauYin2012, author = {Erd{\H{o}}s, L{\'a}szl{\'o} and Yau, Horng-Tzer and Yin, Jun}, title = {Rigidity of eigenvalues of generalized {W}igner matrices}, journal = {Adv. Math.}, volume = {229}, number = {3}, pages = {1435--1515}, year = {2012}, doi = {10.1016/j.aim.2011.12.010}, eprint = {1007.4652}, archivePrefix = {arXiv}, primaryClass = {math-ph} } @misc{LandonSosoe2022, author = {Landon, Benjamin and Sosoe, Philippe}, title = {Almost-optimal bulk regularity conditions in the {CLT} for {W}igner matrices}, howpublished = {Preprint, arXiv:2204.03419}, year = {2022}, eprint = {2204.03419}, archivePrefix = {arXiv}, primaryClass = {math.PR} } % --------------------------------------------------------------------------- % Two-dimensional / other extreme-gap results (context) % --------------------------------------------------------------------------- @article{LopattoMeeker2026, author = {Lopatto, Patrick and Meeker, Matthew}, title = {Smallest gaps between eigenvalues of real {G}aussian matrices}, journal = {Bernoulli}, volume = {32}, number = {1}, pages = {325--349}, year = {2026}, doi = {10.3150/25-BEJ1860}, eprint = {2403.09521}, archivePrefix = {arXiv}, primaryClass = {math.PR} } @misc{LopattoOtto2025, author = {Lopatto, Patrick and Otto, Moritz}, title = {Maximum gap in complex {G}inibre matrices}, howpublished = {Preprint, arXiv:2501.04611}, year = {2025}, eprint = {2501.04611}, archivePrefix = {arXiv}, primaryClass = {math.PR} } @misc{Moreillon2026, author = {Moreillon, Philippe}, title = {Largest bulk gap of the complex {G}inibre ensemble}, howpublished = {Preprint, arXiv:2608.20101}, year = {2026}, eprint = {2608.20101}, archivePrefix = {arXiv}, primaryClass = {math.PR} } @misc{FengMuirhead2026, author = {Feng, Renjie and Muirhead, Stephen}, title = {Poisson approximation of the largest gaps between zeros of a stationary {G}aussian process}, howpublished = {Preprint, arXiv:2605.22146}, year = {2026}, eprint = {2605.22146}, archivePrefix = {arXiv}, primaryClass = {math.PR} } % --------------------------------------------------------------------------- % Sine-kernel gap probabilities (large-gap / hole asymptotics) % --------------------------------------------------------------------------- @article{DesCloizeauxMehta1973, author = {des Cloizeaux, J. and Mehta, M. L.}, title = {Asymptotic behavior of spacing distributions for the eigenvalues of random matrices}, journal = {J. Math. Phys.}, volume = {14}, pages = {1648--1650}, year = {1973}, doi = {10.1063/1.1666239} } @article{Dyson1976, author = {Dyson, F. J.}, title = {Fredholm determinants and inverse scattering problems}, journal = {Comm. Math. Phys.}, volume = {47}, pages = {171--183}, year = {1976}, doi = {10.1007/BF01608375} } @article{Widom1994, author = {Widom, H.}, title = {The asymptotics of a continuous analogue of orthogonal polynomials}, journal = {J. Approx. Theory}, volume = {77}, number = {1}, pages = {51--64}, year = {1994}, doi = {10.1006/jath.1994.1033} } @article{Krasovsky2004, author = {Krasovsky, I. V.}, title = {Gap probability in the spectrum of random matrices and asymptotics of polynomials orthogonal on an arc of the unit circle}, journal = {Int. Math. Res. Not.}, volume = {2004}, number = {25}, pages = {1249--1272}, year = {2004}, doi = {10.1155/S1073792804140221}, eprint = {math/0401258}, archivePrefix = {arXiv}, primaryClass = {math.CA} } @article{Ehrhardt2006, author = {Ehrhardt, Torsten}, title = {{D}yson's constant in the asymptotics of the {F}redholm determinant of the sine kernel}, journal = {Comm. Math. Phys.}, volume = {262}, number = {2}, pages = {317--341}, year = {2006}, doi = {10.1007/s00220-005-1493-4}, eprint = {math/0401205}, archivePrefix = {arXiv}, primaryClass = {math.FA} } @article{DeiftItsKrasovskyZhou2007, author = {Deift, P. and Its, A. and Krasovsky, I. and Zhou, X.}, title = {The {W}idom--{D}yson constant for the gap probability in random matrix theory}, journal = {J. Comput. Appl. Math.}, volume = {202}, number = {1}, pages = {26--47}, year = {2007}, doi = {10.1016/j.cam.2005.12.040}, eprint = {math/0601535}, archivePrefix = {arXiv}, primaryClass = {math.FA} } @article{Ehrhardt2007, author = {Ehrhardt, Torsten}, title = {{D}yson's constants in the asymptotics of the determinants of {W}iener--{H}opf--{H}ankel operators with the sine kernel}, journal = {Comm. Math. Phys.}, volume = {272}, number = {3}, pages = {683--698}, year = {2007}, doi = {10.1007/s00220-007-0239-x}, eprint = {math/0605003}, archivePrefix = {arXiv}, primaryClass = {math.FA} } % --------------------------------------------------------------------------- % Orthogonal / symplectic ensembles: correlation kernels % --------------------------------------------------------------------------- @article{TracyWidom1996, author = {Tracy, Craig A. and Widom, Harold}, title = {On orthogonal and symplectic matrix ensembles}, journal = {Comm. Math. Phys.}, volume = {177}, number = {3}, pages = {727--754}, year = {1996}, doi = {10.1007/BF02099545}, eprint = {solv-int/9509007}, archivePrefix = {arXiv}, primaryClass = {solv-int} } @article{TracyWidom1998, author = {Tracy, Craig A. and Widom, Harold}, title = {Correlation functions, cluster functions and spacing distributions for random matrices}, journal = {J. Stat. Phys.}, volume = {92}, number = {5-6}, pages = {809--835}, year = {1998}, doi = {10.1023/A:1023084324803}, eprint = {solv-int/9804004}, archivePrefix = {arXiv}, primaryClass = {solv-int} } @article{Widom1999, author = {Widom, Harold}, title = {On the relation between orthogonal, symplectic and unitary matrix ensembles}, journal = {J. Stat. Phys.}, volume = {94}, number = {3-4}, pages = {347--363}, year = {1999}, doi = {10.1023/A:1004536018336}, eprint = {solv-int/9804005}, archivePrefix = {arXiv}, primaryClass = {solv-int} } @article{AdlerForresterNagaoVanMoerbeke2000, author = {Adler, M. and Forrester, P. J. and Nagao, T. and van Moerbeke, P.}, title = {Classical skew orthogonal polynomials and random matrices}, journal = {J. Stat. Phys.}, volume = {99}, number = {1-2}, pages = {141--170}, year = {2000}, doi = {10.1023/A:1018644606835}, eprint = {solv-int/9907001}, archivePrefix = {arXiv}, primaryClass = {solv-int} } @incollection{ForresterRains2001, author = {Forrester, P. J. and Rains, E. M.}, title = {Inter-relationships between orthogonal, unitary and symplectic matrix ensembles}, booktitle = {Random Matrix Models and Their Applications}, editor = {Bleher, P. M. and Its, A. R.}, series = {Math. Sci. Res. Inst. Publ.}, volume = {40}, pages = {171--208}, publisher = {Cambridge University Press}, address = {Cambridge}, year = {2001}, eprint = {solv-int/9907008}, archivePrefix = {arXiv}, primaryClass = {solv-int} } @book{Forrester2010, author = {Forrester, Peter J.}, title = {Log-gases and random matrices}, series = {London Mathematical Society Monographs Series}, volume = {34}, publisher = {Princeton University Press}, address = {Princeton, NJ}, year = {2010} } @book{Mehta2004, author = {Mehta, M. L.}, title = {Random matrices}, edition = {Third}, series = {Pure and Applied Mathematics (Amsterdam)}, volume = {142}, publisher = {Elsevier/Academic Press}, address = {Amsterdam}, year = {2004} } % --------------------------------------------------------------------------- % Hermite / orthogonal polynomial asymptotics; universality for beta = 1, 4 % --------------------------------------------------------------------------- @book{Szego1975, author = {Szeg{\H{o}}, G.}, title = {Orthogonal polynomials}, edition = {Fourth}, series = {American Mathematical Society Colloquium Publications}, volume = {XXIII}, publisher = {American Mathematical Society}, address = {Providence, RI}, year = {1975} } @article{PlancherelRotach1929, author = {Plancherel, M. and Rotach, W.}, title = {Sur les valeurs asymptotiques des polynomes d'{H}ermite}, journal = {Comment. Math. Helv.}, volume = {1}, number = {1}, pages = {227--254}, year = {1929}, doi = {10.1007/BF01208365} } @article{DeiftKriecherbauerMcLaughlinVenakidesZhou1999, author = {Deift, P. and Kriecherbauer, T. and McLaughlin, K. T.-R. and Venakides, S. and Zhou, X.}, title = {Strong asymptotics of orthogonal polynomials with respect to exponential weights}, journal = {Comm. Pure Appl. Math.}, volume = {52}, number = {12}, pages = {1491--1552}, year = {1999} } @article{ShiNemesWangWong2023, author = {Shi, Wei and Nemes, Gerg{\H{o}} and Wang, Xiang-Sheng and Wong, Roderick}, title = {Error bounds for the asymptotic expansions of the {H}ermite polynomials}, journal = {Proc. Roy. Soc. Edinburgh Sect. A}, volume = {153}, number = {2}, pages = {417--440}, year = {2023}, doi = {10.1017/prm.2021.90}, eprint = {2109.01054}, archivePrefix = {arXiv}, primaryClass = {math.CA} } @article{DeiftGioev2007a, author = {Deift, Percy and Gioev, Dimitri}, title = {Universality at the edge of the spectrum for unitary, orthogonal and symplectic ensembles of random matrices}, journal = {Comm. Pure Appl. Math.}, volume = {60}, number = {6}, pages = {867--910}, year = {2007}, doi = {10.1002/cpa.20164}, eprint = {math-ph/0507023}, archivePrefix = {arXiv}, primaryClass = {math-ph} } @article{DeiftGioev2007b, author = {Deift, Percy and Gioev, Dimitri}, title = {Universality in random matrix theory for orthogonal and symplectic ensembles}, journal = {Int. Math. Res. Pap. IMRP}, volume = {2007}, pages = {Art. ID rpm004, 116 pp.}, year = {2007}, doi = {10.1093/imrp/rpm004}, eprint = {math-ph/0411075}, archivePrefix = {arXiv}, primaryClass = {math-ph} } @book{DeiftGioev2009, author = {Deift, Percy and Gioev, Dimitri}, title = {Random matrix theory: invariant ensembles and universality}, series = {Courant Lecture Notes in Mathematics}, volume = {18}, publisher = {Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI}, year = {2009} } % --------------------------------------------------------------------------- % Uncertainty principles % --------------------------------------------------------------------------- @article{Nazarov1993, author = {Nazarov, F. L.}, title = {Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type}, journal = {Algebra i Analiz}, volume = {5}, number = {4}, pages = {3--66}, year = {1993}, note = {In Russian; English translation in St. Petersburg Math. J. 5 (1994), no. 4, 663--717} } @article{Jaming2007, author = {Jaming, Philippe}, title = {{N}azarov's uncertainty principles in higher dimension}, journal = {J. Approx. Theory}, volume = {149}, number = {1}, pages = {30--41}, year = {2007}, doi = {10.1016/j.jat.2007.04.005}, eprint = {math/0612367}, archivePrefix = {arXiv}, primaryClass = {math.CA} } @article{AmreinBerthier1977, author = {Amrein, W. O. and Berthier, A. M.}, title = {On support properties of {$L^p$}-functions and their {F}ourier transforms}, journal = {J. Funct. Anal.}, volume = {24}, pages = {258--267}, year = {1977} } % --------------------------------------------------------------------------- % Books % --------------------------------------------------------------------------- @book{AndersonGuionnetZeitouni2010, author = {Anderson, Greg W. and Guionnet, Alice and Zeitouni, Ofer}, title = {An introduction to random matrices}, series = {Cambridge Studies in Advanced Mathematics}, volume = {118}, publisher = {Cambridge University Press}, address = {Cambridge}, year = {2010}, doi = {10.1017/CBO9780511801334} } @book{ErdosYau2017, author = {Erd{\H{o}}s, L{\'a}szl{\'o} and Yau, Horng-Tzer}, title = {A dynamical approach to random matrix theory}, series = {Courant Lecture Notes in Mathematics}, volume = {28}, publisher = {Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI}, year = {2017}, doi = {10.1090/cln/028} } @book{Kallenberg2017, author = {Kallenberg, Olav}, title = {Random measures, theory and applications}, series = {Probability Theory and Stochastic Modelling}, volume = {77}, publisher = {Springer}, address = {Cham}, year = {2017}, doi = {10.1007/978-3-319-41598-7} } @book{Simon2005, author = {Simon, Barry}, title = {Trace ideals and their applications}, edition = {Second}, series = {Mathematical Surveys and Monographs}, volume = {120}, publisher = {American Mathematical Society}, address = {Providence, RI}, year = {2005} } % --------------------------------------------------------------------------- % Pfaffian integration formulas % --------------------------------------------------------------------------- @article{deBruijn1955, author = {de Bruijn, N. G.}, title = {On some multiple integrals involving determinants}, journal = {J. Indian Math. Soc. (N.S.)}, volume = {19}, pages = {133--151}, year = {1955} } @article{BaikRains2001, author = {Baik, Jinho and Rains, Eric M.}, title = {Algebraic aspects of increasing subsequences}, journal = {Duke Math. J.}, volume = {109}, number = {1}, pages = {1--65}, year = {2001}, doi = {10.1215/S0012-7094-01-10911-3}, eprint = {math/9905083}, archivePrefix = {arXiv}, primaryClass = {math.CO} } @article{Sinclair2009, author = {Sinclair, Christopher D.}, title = {Correlation functions for {$\beta = 1$} ensembles of matrices of odd size}, journal = {J. Stat. Phys.}, volume = {136}, number = {1}, pages = {17--33}, year = {2009}, doi = {10.1007/s10955-009-9771-8}, eprint = {0811.1276}, archivePrefix = {arXiv}, primaryClass = {math-ph} } @book{Kallenberg1983, author = {Kallenberg, Olav}, title = {Random measures}, edition = {Third}, publisher = {Akademie-Verlag}, address = {Berlin}, year = {1983} } @article{ErdosYau2015, author = {Erd{\H{o}}s, L{\'a}szl{\'o} and Yau, Horng-Tzer}, title = {Gap universality of generalized {W}igner and $\beta$-ensembles}, journal = {J. Eur. Math. Soc. (JEMS)}, volume = {17}, number = {8}, pages = {1927--2036}, year = {2015}, doi = {10.4171/JEMS/548}, eprint = {1211.3786}, archivePrefix = {arXiv}, primaryClass = {math.PR} } @article{LopattoLuh2021, author = {Lopatto, Patrick and Luh, Kyle}, title = {Tail bounds for gaps between eigenvalues of sparse random matrices}, journal = {Electron. J. Probab.}, volume = {26}, pages = {Paper No. 130}, year = {2021}, doi = {10.1214/21-EJP669}, eprint = {1901.05948}, archivePrefix = {arXiv}, primaryClass = {math.PR} } @article{ForresterMays2009, author = {Forrester, Peter J. and Mays, Anthony}, title = {A method to calculate correlation functions for $\beta = 1$ random matrices of odd size}, journal = {J. Stat. Phys.}, volume = {134}, number = {3}, pages = {443--462}, year = {2009}, doi = {10.1007/s10955-009-9684-6}, eprint = {0809.5116}, archivePrefix = {arXiv}, primaryClass = {math-ph} } @article{FengGotzeYao2024, author = {Feng, Renjie and G{\"o}tze, Friedrich and Yao, Dong}, title = {Smallest gaps between zeros of stationary {G}aussian processes}, journal = {J. Funct. Anal.}, volume = {287}, number = {4}, pages = {Paper No. 110493}, year = {2024}, doi = {10.1016/j.jfa.2024.110493}, eprint = {2607.19660}, archivePrefix = {arXiv}, primaryClass = {math.PR} } @article{FengYao2026, author = {Feng, Renjie and Yao, Dong}, title = {Smallest distances between zeros of {G}aussian analytic functions}, journal = {Adv. Math.}, volume = {496}, pages = {Paper No. 110999}, year = {2026}, doi = {10.1016/j.aim.2026.110999}, eprint = {2604.26316}, archivePrefix = {arXiv}, primaryClass = {math.PR} } @misc{ShiJiang2012, author = {Shi, Dai and Jiang, Yunjiang}, title = {Smallest gaps between eigenvalues of random matrices with complex {G}inibre, {W}ishart and universal unitary ensembles}, howpublished = {Preprint, arXiv:1207.4240}, year = {2012}, eprint = {1207.4240}, archivePrefix = {arXiv}, primaryClass = {math.PR} } @article{Dyson1962, author = {Dyson, Freeman J.}, title = {Statistical theory of the energy levels of complex systems. {I}}, journal = {J. Math. Phys.}, volume = {3}, pages = {140--156}, year = {1962}, doi = {10.1063/1.1703773} } @article{Soshnikov2000, author = {Soshnikov, Alexander}, title = {Determinantal random point fields}, journal = {Russian Math. Surveys}, volume = {55}, number = {5}, pages = {923--975}, year = {2000}, doi = {10.1070/rm2000v055n05ABEH000321}, eprint = {math/0002099}, archivePrefix = {arXiv}, primaryClass = {math.PR} }