{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,21]],"date-time":"2025-09-21T17:45:07Z","timestamp":1758476707823},"reference-count":42,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2018,1,16]],"date-time":"2018-01-16T00:00:00Z","timestamp":1516060800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Found Comput Math"],"published-print":{"date-parts":[[2019,2]]},"DOI":"10.1007\/s10208-018-9376-1","type":"journal-article","created":{"date-parts":[[2018,1,16]],"date-time":"2018-01-16T20:52:42Z","timestamp":1516135962000},"page":"55-82","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Volumes for $${\\mathrm{SL}}_N({\\mathbb {R}})$$ SL N ( R ) , the Selberg Integral and Random Lattices"],"prefix":"10.1007","volume":"19","author":[{"given":"Peter J.","family":"Forrester","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2018,1,16]]},"reference":[{"key":"9376_CR1","first-page":"465210","volume":"45","author":"G Akemann","year":"2012","unstructured":"G.\u00a0Akemann and Z.\u00a0Burda, Universal microscopic correlations for products of independent Ginibre matrices, J. Phys. A 45 (2012), 465210.","journal-title":"J. Phys. A"},{"key":"9376_CR2","doi-asserted-by":"publisher","first-page":"052118","DOI":"10.1103\/PhysRevE.88.052118","volume":"88","author":"G Akemann","year":"2013","unstructured":"G.\u00a0Akemann, J.R. Ipsen, and M.\u00a0Kieburg, Products of rectangular random matrices: Singular values and progressive scattering, Phys. Rev. E 88 (2013), 052118.","journal-title":"Phys. Rev. E"},{"key":"9376_CR3","volume-title":"Lattice basis reduction: an introduction to the LLL algorithm and its applications","author":"MR Bremner","year":"2012","unstructured":"M.R. Bremner, Lattice basis reduction: an introduction to the LLL algorithm and its applications, CRC Press, Boca Raton, FL, 2012."},{"key":"9376_CR4","doi-asserted-by":"publisher","first-page":"1730001","DOI":"10.1142\/S2010326317300017","volume":"6","author":"P Diaconis","year":"2017","unstructured":"P.\u00a0Diaconis and P.J. Forrester, Hurwitz and the origin of random matrix theory in mathematics, Random Matrices: Theory Appl. 6 (2017), 1730001.","journal-title":"Random Matrices: Theory Appl."},{"key":"9376_CR5","doi-asserted-by":"publisher","first-page":"143","DOI":"10.1215\/S0012-7094-93-07107-4","volume":"81","author":"W Duke","year":"1993","unstructured":"W.\u00a0Duke, Z.\u00a0Rudnick and P.\u00a0Sarnak, Density of integer points on affine homogeneous varieties, Duke Math. J. 81 (1993), 143\u2013179.","journal-title":"Duke Math. J."},{"key":"9376_CR6","volume-title":"Acta Numerica","author":"A Edelman","year":"2005","unstructured":"A.\u00a0Edelman and N.\u00a0Raj Rao, Random matrix theory, Acta Numerica (A.\u00a0Iserles, ed.), vol.\u00a014, Cambridge University Press, Cambridge, 2005."},{"key":"9376_CR7","doi-asserted-by":"crossref","DOI":"10.1515\/9781400835416","volume-title":"Log-gases and random matrices","author":"PJ Forrester","year":"2010","unstructured":"P.J. Forrester, Log-gases and random matrices, Princeton University Press, Princeton, NJ, 2010."},{"key":"9376_CR8","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1007\/s00220-004-1121-8","volume":"250","author":"PJ Forrester","year":"2004","unstructured":"P.J. Forrester and J.P. Keating, Singularity dominated strong fluctuations for some random matrix averages, Commun. Math. Phys. 250 (2004), 119\u2013131.","journal-title":"Commun. Math. Phys."},{"key":"9376_CR9","doi-asserted-by":"publisher","first-page":"771","DOI":"10.1007\/s00220-011-1305-y","volume":"309","author":"PJ Forrester","year":"2012","unstructured":"P.J. Forrester and E.M. Rains, A Fuchsian matrix differential equation for Selberg correlation integrals, Commun. Math. Phys. 309, (2012), 771\u2013792.","journal-title":"Commun. Math. Phys."},{"key":"9376_CR10","doi-asserted-by":"publisher","first-page":"489","DOI":"10.1090\/S0273-0979-08-01221-4","volume":"45","author":"PJ Forrester","year":"2008","unstructured":"P.J. Forrester and S.O. Warnaar, The importance of the Selberg integral, Bull. Am. Math. Soc. 45 (2008), 489\u2013534.","journal-title":"Bull. Am. Math. Soc."},{"key":"9376_CR11","unstructured":"E. Fuchs and I. Rivin, Generic thinness in finitely generated subgroups of\u00a0 $$SL{}_n(\\mathbb{Z})$$ S L n ( Z ) , arXiv:1506.01735 ."},{"key":"9376_CR12","doi-asserted-by":"publisher","first-page":"165","DOI":"10.1515\/form.2003.009","volume":"15","author":"D Goldstein","year":"2003","unstructured":"D.\u00a0Goldstein and A.Mayer, On the equidistribution of Hecke points, Forum Math. 15 (2003), 165\u2013189.","journal-title":"Forum Math."},{"key":"9376_CR13","first-page":"283","volume":"51","author":"A Hardy","year":"2015","unstructured":"A.\u00a0Hardy, Average characteristic polynomials of determinant point processes, Ann. L\u2019Institut Henri Poincar\u00e9 \u2013 Prob.\u00a0et Stat. 51 (2015), 283\u2013303.","journal-title":"Stat."},{"key":"9376_CR14","unstructured":"A.\u00a0Hurwitz, \u00dcber die Erzeugung der Invarianten durch Integration, Nachr. Ges. Wiss. G\u00f6ttingen (1897), 71\u201390."},{"key":"9376_CR15","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1017\/S0013091500011731","volume":"15","author":"H Jack","year":"1967","unstructured":"H.\u00a0Jack, The asymptotic value of the volume of a certain set of matrices, Proc. Edinburgh Math. Soc. 15 (1967), 209\u2013213.","journal-title":"Proc. Edinburgh Math. Soc."},{"key":"9376_CR16","doi-asserted-by":"publisher","first-page":"213","DOI":"10.1017\/S0305004100033922","volume":"55","author":"H Jack","year":"1959","unstructured":"H.\u00a0Jack and A.M. Macbeath, The volume of a certain set of matrices, Math. Proc. Camb. Phil. Soc. 55 (1959), 213\u2013223.","journal-title":"Math. Proc. Camb. Phil. Soc."},{"key":"9376_CR17","doi-asserted-by":"crossref","unstructured":"M.\u00a0Kieburg, A.B.J.\u00a0Kuijlaars and D.\u00a0Stivigny Singular value statistics of matrix products with truncated unitary matrices, Int. Math. Research Notices 2016 (2016), 3392\u20133424.","DOI":"10.1093\/imrn\/rnv242"},{"key":"9376_CR18","unstructured":"S.\u00a0Kim, On the shape of a high-dimensional random lattice, Ph.D. thesis, Stanford University, 2015."},{"key":"9376_CR19","doi-asserted-by":"publisher","first-page":"1117","DOI":"10.1007\/s00209-015-1580-y","volume":"282","author":"S Kim","year":"2016","unstructured":"S.\u00a0Kim, On the distribution of lengths of short vectors in a random lattice, Mathematische Zeitschrift 282 (2016), 1117\u20131126.","journal-title":"Mathematische Zeitschrift"},{"key":"9376_CR20","doi-asserted-by":"publisher","first-page":"565","DOI":"10.1017\/S0305004100030656","volume":"51","author":"AM Macbeath","year":"1955","unstructured":"A.M. Macbeath and C.A. Rogers, A modified form of Siegel\u2019s mean value theorem, Math.\u00a0Proc. Camb. Phil. Soc. 51 (1955), 565\u2013576.","journal-title":"Math. Proc. Camb. Phil. Soc."},{"key":"9376_CR21","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1017\/S0305004100033302","volume":"54","author":"AM Macbeath","year":"1958","unstructured":"A.M. Macbeath and C.A. Rogers, Siegel\u2019s mean value theorem in the geometry of numbers, Math.\u00a0Proc. Camb. Phil. Soc. 54 (1958), 139\u2013151.","journal-title":"Math. Proc. Camb. Phil. Soc."},{"key":"9376_CR22","doi-asserted-by":"publisher","first-page":"1127","DOI":"10.1017\/S0143385700000626","volume":"20","author":"J Marklof","year":"2000","unstructured":"J. Marklof, The $$n$$ n -point correlations between value of a linear form, Ergod. Th. & Dyn. Systems 20 (2000), 1127\u20131172.","journal-title":"Ergod. Th. & Dyn. Systems"},{"key":"9376_CR23","doi-asserted-by":"publisher","first-page":"1413","DOI":"10.1088\/0951-7715\/21\/7\/001","volume":"21","author":"J Marklof","year":"2008","unstructured":"J. Marklof and A. Str\u00f6mbergsson, Kinetic transport in the two-dimensional periodic Lorentz gas, Nonlinearity 21 (2008), 1413\u20131422.","journal-title":"Nonlinearity"},{"key":"9376_CR24","doi-asserted-by":"publisher","first-page":"1949","DOI":"10.4007\/annals.2010.172.1949","volume":"172","author":"J Marklof","year":"2010","unstructured":"J. Marklof and A. Str\u00f6mbergsson, The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems, Ann. Math. 172 (2010): 1949\u20132033.","journal-title":"Ann. Math."},{"key":"9376_CR25","doi-asserted-by":"publisher","first-page":"560","DOI":"10.1007\/s00039-011-0116-9","volume":"21","author":"J Marklof","year":"2011","unstructured":"J. Marklof and A. Str\u00f6mbergsson, The periodic Lorentz gas in the Boltzmann-Grad limit: asymptotic estimates, Geometric and Functional Analysis 21 (2011) 560\u2013647.","journal-title":"Geometric and Functional Analysis"},{"key":"9376_CR26","doi-asserted-by":"publisher","first-page":"429","DOI":"10.1007\/s00493-013-2820-6","volume":"33","author":"J Marklof","year":"2013","unstructured":"J. Marklof and A. Str\u00f6mbergsson, Diameters of random circulant graphs, Combinatorica 33 (2013) 429-466.","journal-title":"Combinatorica"},{"key":"9376_CR27","doi-asserted-by":"publisher","DOI":"10.1142\/3438","volume-title":"Jacobians of matrix transformations and functions of matrix arguments","author":"AM Mathai","year":"1997","unstructured":"A.M. Mathai, Jacobians of matrix transformations and functions of matrix arguments, World Scientific, Singapore, 1997."},{"key":"9376_CR28","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1515\/crll.1905.129.220","volume":"129","author":"H Minkowski","year":"1905","unstructured":"H.\u00a0Minkowski, Diskontinuit\u00e4tsbereich f\u00fcr arithmetische \u00e4quivalenz, J. Reine Angew. Math. 129 (1905), 220\u2013274.","journal-title":"J. Reine Angew. Math."},{"key":"9376_CR29","doi-asserted-by":"publisher","DOI":"10.1002\/9780470316559","volume-title":"Aspects of multivariate statistical theory","author":"RJ Muirhead","year":"1982","unstructured":"R.J. Muirhead, Aspects of multivariate statistical theory, Wiley, New York, 1982."},{"key":"9376_CR30","unstructured":"P.Q. Nguyen and D.\u00a0Stehl\u00e9, Low-dimensional lattice basis reduction revisited, Algorithmic number theory, Lecture notes in computer science, vol. 3076, Springer Berlin Heidelberg, 2001, pp.\u00a0338\u2013357."},{"key":"9376_CR31","doi-asserted-by":"publisher","first-page":"207","DOI":"10.1007\/s10955-008-9539-6","volume":"132","author":"G Parisi","year":"2008","unstructured":"G. Parisi, On the most compact regular lattices in large dimensions: A statistical mechanical approach, J. Stat. Phys. 132 (2008), 207\u2013234.","journal-title":"J. Stat. Phys."},{"key":"9376_CR32","doi-asserted-by":"publisher","first-page":"783","DOI":"10.1090\/mcom\/2986","volume":"85","author":"I Rivin","year":"2016","unstructured":"I.\u00a0Rivin, How to pick a random integer matrix? (and other questions), Math. Computation 85 (2016), 783\u2013797.","journal-title":"Math. Computation"},{"key":"9376_CR33","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1098\/rsta.1955.0015","volume":"248","author":"CA Rogers","year":"1955","unstructured":"C.A.\u00a0Rogers, The moments of the number of points of a lattice in a bounded set, Phil. Trans. R. Soc. Lond. A 248 (1955), 225\u2013251.","journal-title":"Phil. Trans. R. Soc. Lond. A"},{"key":"9376_CR34","first-page":"71","volume":"24","author":"A Selberg","year":"1944","unstructured":"A.\u00a0Selberg, Bemerkninger om et multipelt integral, Norsk. Mat. Tidsskr. 24 (1944), 71\u201378.","journal-title":"Norsk. Mat. Tidsskr."},{"key":"9376_CR35","doi-asserted-by":"crossref","unstructured":"I.\u00a0Semaev, A 3-dimensional lattice reduction algorithm, Proc.\u00a0of CALC \u201901 (P.\u00a0Huber and M.\u00a0Rosenblatt, eds.), Lecture notes in computer science, vol. 2146, Springer-Verlag, 2001, pp.\u00a0183\u2013193.","DOI":"10.1007\/3-540-44670-2_13"},{"key":"9376_CR36","doi-asserted-by":"publisher","first-page":"73","DOI":"10.17323\/1609-4514-2001-1-1-73-89","volume":"1","author":"S Shlosman","year":"2001","unstructured":"S.\u00a0Shlosman and M.A.\u00a0Tsfasman, Random lattices and random sphere packings: typical properties, Moscow Math. Journal 1 (2001), 73\u201389.","journal-title":"Moscow Math. Journal"},{"key":"9376_CR37","doi-asserted-by":"publisher","first-page":"340","DOI":"10.2307\/1969027","volume":"46","author":"CL Siegel","year":"1945","unstructured":"C.L. Siegel, A mean value theorem in geometry of numbers, Ann. Math. 46 (1945), 340\u2013347.","journal-title":"Ann. Math."},{"key":"9376_CR38","doi-asserted-by":"publisher","first-page":"945","DOI":"10.1007\/s00209-010-0772-8","volume":"269","author":"A S\u00f6dergren","year":"2011","unstructured":"A.\u00a0S\u00f6dergren, On the Poisson distribution of lengths of lattice vectors in a random lattice, Mathematische Zeitschrift 269 (2011), 945\u2013954.","journal-title":"Mathematische Zeitschrift"},{"key":"9376_CR39","doi-asserted-by":"crossref","unstructured":"A.\u00a0S\u00f6dergren, On the distribution of angles between the\u00a0 $$N$$ N \u00a0shortest vectors in a random lattice, J. London Math. Soc. 84 (2011), 749\u2013764.","DOI":"10.1112\/jlms\/jdr032"},{"key":"9376_CR40","doi-asserted-by":"publisher","first-page":"41","DOI":"10.4064\/aa118-1-4","volume":"118","author":"A Str\u00f6mbergsson","year":"2005","unstructured":"A. Str\u00f6mbergsson and A. Venkatesh, Small solutions to linear congruences and Hecke equidistribution, Acta Arith., 118 (2005), 41-78.","journal-title":"Acta Arith."},{"key":"9376_CR41","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-3820-1","volume-title":"Harmonic analysis on symmetric spaces and applications","author":"A Terras","year":"1988","unstructured":"A.\u00a0Terras, Harmonic analysis on symmetric spaces and applications, vol.\u00a02, Springer-Verlag, Birlin, 1988."},{"key":"9376_CR42","doi-asserted-by":"publisher","first-page":"7111","DOI":"10.1088\/0305-4470\/34\/35\/335","volume":"34","author":"K Zyczkowski","year":"2001","unstructured":"K.\u00a0Zyczkowski and H.-J. Sommers, Induced measures in the space of mixed quantum states, J.\u00a0Phys. A 34 (2001), 7111\u20137125.","journal-title":"J. Phys. A"}],"container-title":["Foundations of Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10208-018-9376-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-018-9376-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-018-9376-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,10,9]],"date-time":"2019-10-09T12:10:53Z","timestamp":1570623053000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10208-018-9376-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1,16]]},"references-count":42,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2019,2]]}},"alternative-id":["9376"],"URL":"https:\/\/doi.org\/10.1007\/s10208-018-9376-1","relation":{},"ISSN":["1615-3375","1615-3383"],"issn-type":[{"value":"1615-3375","type":"print"},{"value":"1615-3383","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1,16]]},"assertion":[{"value":"21 April 2017","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 April 2017","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 October 2017","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 January 2018","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}