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Volumes for \({\mathrm{SL}}_N({\mathbb {R}})\), the Selberg Integral and Random Lattices

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Abstract

There is a natural left and right invariant Haar measure associated with the matrix groups GL\({}_N(\mathbb {R})\) and SL\({}_N(\mathbb {R})\) due to Siegel. For the associated volume to be finite it is necessary to truncate the groups by imposing a bound on the norm, or in the case of SL\({}_N(\mathbb {R})\), by restricting to a fundamental domain. We compute the asymptotic volumes associated with the Haar measure for GL\({}_N(\mathbb {R})\) and SL\({}_N(\mathbb {R})\) matrices in the case that the singular values lie between \(R_1\) and \(1/R_2\) in the former, and that the 2-norm, or alternatively the Frobenius norm, is bounded by R in the latter. By a result of Duke, Rudnick and Sarnak, such asymptotic formulas in the case of SL\({}_N(\mathbb {R})\) imply an asymptotic counting formula for matrices in SL\({}_N(\mathbb {Z})\). We discuss too the sampling of SL\({}_N(\mathbb {R})\) matrices from the truncated sets. By then using lattice reduction to a fundamental domain, we obtain histograms approximating the probability density functions of the lengths and pairwise angles of shortest length bases vectors in the case \(N=2\) and 3, or equivalently of shortest linearly independent vectors in the corresponding random lattice. In the case \(N=2\) these distributions are evaluated explicitly.

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Acknowledgements

This research Project is part of the programme of study supported by the ARC Centre of Excellence for Mathematical and Statistical Frontiers. Additional partial support from the Australian Research Council through the Grant DP140102613 is also acknowledged. Helpful and appreciated remarks on an earlier draft of this work have been made by J. Marklof and A. Strömbergsson, with the latter being responsible for the comments in Remark 4.6 relating to [40]. Useful comments of a referee are also to be thanked.

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  1. Department of Mathematics and Statistics, ARC Centre of Excellence for Mathematical and Statistical Frontiers, The University of Melbourne, Melbourne, VIC, 3010, Australia

    Peter J. Forrester

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Correspondence to Peter J. Forrester.

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Communicated by Ian Sloan.

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Forrester, P.J. Volumes for \({\mathrm{SL}}_N({\mathbb {R}})\), the Selberg Integral and Random Lattices. Found Comput Math 19, 55–82 (2019). https://doi.org/10.1007/s10208-018-9376-1

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