The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems

Abstract

The periodic Lorentz gas describes the dynamics of a point particle in a periodic array of spherical scatterers, and is one of the fundamental models for chaotic diffusion. In the present paper we investigate the Boltzmann-Grad limit, where the radius of each scatterer tends to zero, and prove the existence of a limiting distribution for the free path length. We also discuss related problems, such as the statistical distribution of directions of lattice points that are visible from a fixed position.

  • [Balint07] Go to document P. Bálint and I. P. Tóth, "Exponential decay of correlations in multi-dimensional dispersing billiards," Ann. Henri Poincaré, vol. 9, iss. 7, pp. 1309-1369, 2008.
    @article {Balint07, MRKEY = {2453251},
      AUTHOR = {B{á}lint, P{é}ter and T{ó}th, Imre P{é}ter},
      TITLE = {Exponential decay of correlations in multi-dimensional dispersing billiards},
      JOURNAL = {Ann. Henri Poincaré},
      FJOURNAL = {Annales Henri Poincaré. A Journal of Theoretical and Mathematical Physics},
      VOLUME = {9},
      YEAR = {2008},
      NUMBER = {7},
      PAGES = {1309--1369},
      ISSN = {1424-0637},
      MRCLASS = {37D50 (37A25 37A60)},
      MRNUMBER = {2010a:37069},
      MRREVIEWER = {Nikolai Chernov},
      DOI = {10.1007/s00023-008-0389-1},
      ZBLNUMBER = {1162.37018},
      }
  • [Bleher92] Go to document P. M. Bleher, "Statistical properties of two-dimensional periodic Lorentz gas with infinite horizon," J. Statist. Phys., vol. 66, iss. 1-2, pp. 315-373, 1992.
    @article {Bleher92, MRKEY = {1149489},
      AUTHOR = {Bleher, P. M.},
      TITLE = {Statistical properties of two-dimensional periodic {L}orentz gas with infinite horizon},
      JOURNAL = {J. Statist. Phys.},
      FJOURNAL = {Journal of Statistical Physics},
      VOLUME = {66},
      YEAR = {1992},
      NUMBER = {1-2},
      PAGES = {315--373},
      ISSN = {0022-4715},
      CODEN = {JSTPSB},
      MRCLASS = {82C21 (82C40)},
      MRNUMBER = {93b:82052},
      MRREVIEWER = {Min Ping Qian},
      ZBLNUMBER = {0925.82147},
      URL={http://www.springerlink.com/content/hk3613380hg27436/},
      }
  • [Boca00] Go to document F. P. Boca, C. Cobeli, and A. Zaharescu, "Distribution of lattice points visible from the origin," Comm. Math. Phys., vol. 213, iss. 2, pp. 433-470, 2000.
    @article {Boca00, MRKEY = {1785463},
      AUTHOR = {Boca, Florin P. and Cobeli, Cristian and Zaharescu, Alexandru},
      TITLE = {Distribution of lattice points visible from the origin},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {213},
      YEAR = {2000},
      NUMBER = {2},
      PAGES = {433--470},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {11P21 (11J54)},
      MRNUMBER = {2001j:11094},
      MRREVIEWER = {Ze{é}v Rudnick},
      DOI = {10.1007/s002200000250},
      ZBLNUMBER = {0989.11049},
      }
  • [Boca03] Go to document F. P. Boca, R. N. Gologan, and A. Zaharescu, "The statistics of the trajectory of a certain billiard in a flat two-torus," Comm. Math. Phys., vol. 240, iss. 1-2, pp. 53-73, 2003.
    @article {Boca03, MRKEY = {2004979},
      AUTHOR = {Boca, Florin P. and Gologan, Radu N. and Zaharescu, Alexandru},
      TITLE = {The statistics of the trajectory of a certain billiard in a flat two-torus},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {240},
      YEAR = {2003},
      NUMBER = {1-2},
      PAGES = {53--73},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {37A60 (37N20 82B05)},
      MRNUMBER = {2004f:37007},
      MRREVIEWER = {Serge E. Troubetzkoy},
      DOI = {10.1007/s00220-003-0907-4},
      ZBLNUMBER = {1078.37006},
      }
  • [Boca05] Go to document F. P. Boca and A. Zaharescu, "The correlations of Farey fractions," J. London Math. Soc., vol. 72, iss. 1, pp. 25-39, 2005.
    @article {Boca05, MRKEY = {2145726},
      AUTHOR = {Boca, Florin P. and Zaharescu, Alexandru},
      TITLE = {The correlations of {F}arey fractions},
      JOURNAL = {J. London Math. Soc.},
      FJOURNAL = {Journal of the London Mathematical Society. Second Series},
      VOLUME = {72},
      YEAR = {2005},
      NUMBER = {1},
      PAGES = {25--39},
      ISSN = {0024-6107},
      CODEN = {JLMSAK},
      MRCLASS = {11J71 (11B57 11K36)},
      MRNUMBER = {2006h:11086},
      MRREVIEWER = {Dmitry Y. Kleinbock},
      DOI = {10.1112/S0024610705006629},
      ZBLNUMBER = {1089.11037},
      }
  • [Boca06] Go to document F. P. Boca and A. Zaharescu, "On the correlations of directions in the Euclidean plane," Trans. Amer. Math. Soc., vol. 358, iss. 4, pp. 1797-1825, 2006.
    @article {Boca06, MRKEY = {2186997},
      AUTHOR = {Boca, Florin P. and Zaharescu, Alexandru},
      TITLE = {On the correlations of directions in the {E}uclidean plane},
      JOURNAL = {Trans. Amer. Math. Soc.},
      FJOURNAL = {Transactions of the American Mathematical Society},
      VOLUME = {358},
      YEAR = {2006},
      NUMBER = {4},
      PAGES = {1797--1825},
      ISSN = {0002-9947},
      CODEN = {TAMTAM},
      MRCLASS = {11J71},
      MRNUMBER = {2007e:11077},
      MRREVIEWER = {Takao Komatsu},
      DOI = {10.1090/S0002-9947-05-03783-9},
      ZBLNUMBER = {1154.11022},
      }
  • [Boca07] Go to document F. P. Boca and A. Zaharescu, "The distribution of the free path lengths in the periodic two-dimensional Lorentz gas in the small-scatterer limit," Comm. Math. Phys., vol. 269, iss. 2, pp. 425-471, 2007.
    @article {Boca07, MRKEY = {2274553},
      AUTHOR = {Boca, Florin P. and Zaharescu, Alexandru},
      TITLE = {The distribution of the free path lengths in the periodic two-dimensional {L}orentz gas in the small-scatterer limit},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {269},
      YEAR = {2007},
      NUMBER = {2},
      PAGES = {425--471},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {37A25 (11B57 37A60 37D50 82C05 82C40)},
      MRNUMBER = {2008f:37013},
      MRREVIEWER = {Nikolai Chernov},
      DOI = {10.1007/s00220-006-0137-7},
      ZBLNUMBER = {1143.37002},
      }
  • [Boldrighini83] Go to document C. Boldrighini, L. A. Bunimovich, and Y. G. Sinai, "On the Boltzmann equation for the Lorentz gas," J. Statist. Phys., vol. 32, iss. 3, pp. 477-501, 1983.
    @article {Boldrighini83, MRKEY = {725107},
      AUTHOR = {Boldrighini, C. and Bunimovich, L. A. and Sinai, Ya. G.},
      TITLE = {On the {B}oltzmann equation for the {L}orentz gas},
      JOURNAL = {J. Statist. Phys.},
      FJOURNAL = {Journal of Statistical Physics},
      VOLUME = {32},
      YEAR = {1983},
      NUMBER = {3},
      PAGES = {477--501},
      ISSN = {0022-4715},
      CODEN = {JSTPSB},
      MRCLASS = {82A40 (60K35 82A05)},
      MRNUMBER = {85h:82025},
      MRREVIEWER = {Carlo Cercignani},
      ZBLNUMBER = {0583.76092},
      URL={http://www.springerlink.com/content/h601778507103m40/},
      }
  • [Bourgain98] Go to document J. Bourgain, F. Golse, and B. Wennberg, "On the distribution of free path lengths for the periodic Lorentz gas," Comm. Math. Phys., vol. 190, iss. 3, pp. 491-508, 1998.
    @article {Bourgain98, MRKEY = {1600299},
      AUTHOR = {Bourgain, Jean and Golse, Fran{ç}ois and Wennberg, Bernt},
      TITLE = {On the distribution of free path lengths for the periodic {L}orentz gas},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {190},
      YEAR = {1998},
      NUMBER = {3},
      PAGES = {491--508},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {82C40 (58F40)},
      MRNUMBER = {98k:82146},
      MRREVIEWER = {Nikolai Chernov},
      DOI = {10.1007/s002200050249},
      ZBLNUMBER = {0910.60082},
      }
  • [Bunimovich80] Go to document L. A. Bunimovich and Y. G. Sinai, "Statistical properties of Lorentz gas with periodic configuration of scatterers," Comm. Math. Phys., vol. 78, iss. 4, pp. 479-497, 1980/81.
    @article {Bunimovich80, MRKEY = {606459},
      AUTHOR = {Bunimovich, L. A. and Sinai, Ya. G.},
      TITLE = {Statistical properties of {L}orentz gas with periodic configuration of scatterers},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {78},
      YEAR = {1980/81},
      NUMBER = {4},
      PAGES = {479--497},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {82A05 (58F11 58F13 60F17 60K35)},
      MRNUMBER = {82m:82007},
      MRREVIEWER = {Sheldon Goldstein},
      URL = {http://projecteuclid.org/getRecord?id=euclid.cmp/1103908786},
      ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
      ZBLNUMBER = {0459.60099},
      }
  • [Caglioti03] Go to document E. Caglioti and F. Golse, "On the distribution of free path lengths for the periodic Lorentz gas. III," Comm. Math. Phys., vol. 236, iss. 2, pp. 199-221, 2003.
    @article {Caglioti03, MRKEY = {1981109},
      AUTHOR = {Caglioti, Emanuele and Golse, Fran{ç}ois},
      TITLE = {On the distribution of free path lengths for the periodic {L}orentz gas. {III}},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {236},
      YEAR = {2003},
      NUMBER = {2},
      PAGES = {199--221},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {82C05 (37A60 37D50 82C40)},
      MRNUMBER = {2004h:82060},
      MRREVIEWER = {Nikolai Chernov},
      DOI = {10.1007/s00220-003-0825-5},
      ZBLNUMBER = {1041.82016},
      }
  • [Chernov94] Go to document N. I. Chernov, "Statistical properties of the periodic Lorentz gas. Multidimensional case," J. Statist. Phys., vol. 74, iss. 1-2, pp. 11-53, 1994.
    @article {Chernov94, MRKEY = {1257815},
      AUTHOR = {Chernov, N. I.},
      TITLE = {Statistical properties of the periodic {L}orentz gas. {M}ultidimensional case},
      JOURNAL = {J. Statist. Phys.},
      FJOURNAL = {Journal of Statistical Physics},
      VOLUME = {74},
      YEAR = {1994},
      NUMBER = {1-2},
      PAGES = {11--53},
      ISSN = {0022-4715},
      CODEN = {JSTPSB},
      MRCLASS = {58F11 (58F03 58F15 82C40)},
      MRNUMBER = {95d:58083},
      MRREVIEWER = {Andr{á}s Kr{á}mli},
      ZBLNUMBER = {0946.37500},
      URL={http://www.springerlink.com/content/k13l38756q0267n1/},
      }
  • [dolgopyat] Go to document D. Dolgopyat, D. Szász, and T. Varjú, "Recurrence properties of planar Lorentz process," Duke Math. J., vol. 142, iss. 2, pp. 241-281, 2008.
    @article {dolgopyat, MRKEY = {2401621},
      AUTHOR = {Dolgopyat, Dmitry and Sz{á}sz, Domokos and Varj{ú},
      Tam{á}s},
      TITLE = {Recurrence properties of planar {L}orentz process},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {142},
      YEAR = {2008},
      NUMBER = {2},
      PAGES = {241--281},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {37D50 (37A50 37C30 60F05 60K40)},
      MRNUMBER = {2009h:37074},
      MRREVIEWER = {Nikolai Chernov},
      DOI = {10.1215/00127094-2008-006},
      ZBLNUMBER = {1136.37022},
      }
  • [Elkies04] Go to document N. D. Elkies and C. T. McMullen, "Gaps in ${\sqrt n}\bmod 1$ and ergodic theory," Duke Math. J., vol. 123, iss. 1, pp. 95-139, 2004.
    @article {Elkies04, MRKEY = {2060024},
      AUTHOR = {Elkies, Noam D. and McMullen, Curtis T.},
      TITLE = {Gaps in {${\sqrt n}\bmod 1$} and ergodic theory},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {123},
      YEAR = {2004},
      NUMBER = {1},
      PAGES = {95--139},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {11J71 (22E40 37A17 37A25 37A45)},
      MRNUMBER = {2005f:11143},
      MRREVIEWER = {Ze{é}v Rudnick},
      DOI = {10.1215/S0012-7094-04-12314-0},
      ZBLNUMBER = {1063.11020},
      }
  • [Eskin93] Go to document A. Eskin and C. McMullen, "Mixing, counting, and equidistribution in Lie groups," Duke Math. J., vol. 71, iss. 1, pp. 181-209, 1993.
    @article {Eskin93, MRKEY = {1230290},
      AUTHOR = {Eskin, Alex and McMullen, Curt},
      TITLE = {Mixing, counting, and equidistribution in {L}ie groups},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {71},
      YEAR = {1993},
      NUMBER = {1},
      PAGES = {181--209},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {22E40 (57S30 58F17)},
      MRNUMBER = {95b:22025},
      MRREVIEWER = {Nimish A. Shah},
      DOI = {10.1215/S0012-7094-93-07108-6},
      ZBLNUMBER = {0798.11025},
      }
  • [Gallavotti69] Go to document G. Gallavotti, "Divergences and approach to equilibrium in the Lorentz and the wind-tree-models," Phys. Rev., vol. 185, pp. 308-322, 1969.
    @article{Gallavotti69,
      author={Gallavotti, G.},
      TITLE={Divergences and approach to equilibrium in the Lorentz and the wind-tree-models},
      JOURNAL={Phys. Rev.},
      VOLUME={185},
      YEAR={1969},
      PAGES={308--322},
      URL={http://prola.aps.org/abstract/PR/v185/i1/p308_1},
      }
  • [Golse06] F. Golse, "The periodic Lorentz gas in the Boltzmann-Grad limit," in International Congress of Mathematicians. Vol. III, Eur. Math. Soc., Zürich, 2006, pp. 183-201.
    @incollection {Golse06, MRKEY = {2275676},
      AUTHOR = {Golse, Fran{ç}ois},
      TITLE = {The periodic {L}orentz gas in the {B}oltzmann-{G}rad limit},
      BOOKTITLE = {International {C}ongress of {M}athematicians. {V}ol. {\rm III}},
      PAGES = {183--201},
      PUBLISHER = {Eur. Math. Soc., Zürich},
      YEAR = {2006},
      MRCLASS = {82C40 (37A60 37D50)},
      MRNUMBER = {2007k:82112},
      MRREVIEWER = {Silvia Caprino},
      ZBLNUMBER = {1109.82020},
      }
  • [Golse07] Go to document F. Golse, "On the periodic Lorentz gas and the Lorentz kinetic equation," Ann. Fac. Sci. Toulouse Math., vol. 17, iss. 4, pp. 735-749, 2008.
    @article {Golse07, MRKEY = {2499853},
      AUTHOR = {Golse, Fran{ç}ois},
      TITLE = {On the periodic {L}orentz gas and the {L}orentz kinetic equation},
      JOURNAL = {Ann. Fac. Sci. Toulouse Math.},
      FJOURNAL = {Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série 6},
      VOLUME = {17},
      YEAR = {2008},
      NUMBER = {4},
      PAGES = {735--749},
      ISSN = {0240-2963},
      MRCLASS = {82C40},
      MRNUMBER = {2010e:82080},
      URL = {http://afst.cedram.org/item?id=AFST_2008_6_17_4_735_0},
      ZBLNUMBER = {1166.82304},
      }
  • [Golse00] Go to document F. Golse and B. Wennberg, "On the distribution of free path lengths for the periodic Lorentz gas. II," M2AN Math. Model. Numer. Anal., vol. 34, iss. 6, pp. 1151-1163, 2000.
    @article {Golse00, MRKEY = {1812731},
      AUTHOR = {Golse, Fran{ç}ois and Wennberg, Bernt},
      TITLE = {On the distribution of free path lengths for the periodic {L}orentz gas. {II}},
      JOURNAL = {M2AN Math. Model. Numer. Anal.},
      FJOURNAL = {M2AN. Mathematical Modelling and Numerical Analysis},
      VOLUME = {34},
      YEAR = {2000},
      NUMBER = {6},
      PAGES = {1151--1163},
      ISSN = {0764-583X},
      MRCLASS = {82C40 (35F15 37D50)},
      MRNUMBER = {2002a:82092},
      MRREVIEWER = {Nikolai Chernov},
      DOI = {10.1051/m2an:2000121},
      ZBLNUMBER = {1006.82025},
      }
  • [Lorentz] H. Lorentz, "Le mouvement des électrons dans les métaux," Arch. Néerl, vol. 10, pp. 336-371, 1905.
    @article{Lorentz,
      author={Lorentz, H.},
      TITLE={Le mouvement des électrons dans les métaux},
      JOURNAL={Arch. Néerl},
      VOLUME={10},
      YEAR={1905},
      PAGES={336--371},
      }
  • [Marklof00] Go to document J. Marklof, "The $n$-point correlations between values of a linear form," Ergodic Theory Dynam. Systems, vol. 20, iss. 4, pp. 1127-1172, 2000.
    @article {Marklof00, MRKEY = {1779397},
      AUTHOR = {Marklof, Jens},
      TITLE = {The {$n$}-point correlations between values of a linear form},
      JOURNAL = {Ergodic Theory Dynam. Systems},
      FJOURNAL = {Ergodic Theory and Dynamical Systems},
      VOLUME = {20},
      YEAR = {2000},
      NUMBER = {4},
      PAGES = {1127--1172},
      ISSN = {0143-3857},
      MRCLASS = {11J13 (37A17 37A45)},
      MRNUMBER = {2001m:11112},
      MRREVIEWER = {Dmitry Y. Kleinbock},
      DOI = {10.1017/S0143385700000626},
      ZBLNUMBER = {0967.60021},
      }
  • [Marklof07] J. Marklof, "Distribution modulo one and Ratner’s theorem," in Equidistribution in Number Theory, an Introduction, New York: Springer-Verlag, 2007, vol. 237, pp. 217-244.
    @incollection {Marklof07, MRKEY = {2290501},
      AUTHOR = {Marklof, Jens},
      TITLE = {Distribution modulo one and {R}atner's theorem},
      BOOKTITLE = {Equidistribution in Number Theory, an Introduction},
      SERIES = {NATO Sci. Ser. II Math. Phys. Chem.},
      VOLUME = {237},
      PAGES = {217--244},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2007},
      MRCLASS = {11K31 (11J71 11K06 28D15 37A45)},
      MRNUMBER = {2007k:11118},
      MRREVIEWER = {Thomas Ward},
      ZBLNUMBER = {1181.11048},
      }
  • [partII] J. Marklof and A. Strömbergsson, "The Boltzmann-Grad limit of the periodic Lorentz gas."
    @article{partII,
      author = {Marklof, Jens and Strömbergsson, A},
      TITLE={The Boltzmann-Grad limit of the periodic Lorentz gas},
      ARXIV={0801.0612},
      NOTE={{\it Ann. of Math.},
      to appear},
      }
  • [partIII] Go to document J. Marklof and A. Strömbergsson, "Kinetic transport in the two-dimensional periodic Lorentz gas," Nonlinearity, vol. 21, pp. 1413-1422, 2008.
    @article{partIII,
      author = {Marklof, Jens and Strömbergsson, A},
      TITLE={Kinetic transport in the two-dimensional periodic Lorentz gas},
      JOURNAL={Nonlinearity},
      VOLUME={21},
      YEAR={2008},
      PAGES={1413--1422},
      URL={http://iopscience.iop.org/0951-7715/21/7/001/},
      MRNUMBER={2009j:82029},
      ZBLNUMBER={1153.82020},
      }
  • [partIV] J. Marklof and A. Strömbergsson, The periodic Lorentz gas in the Boltzmann-Grad limit: Asymptotic estimates.
    @misc{partIV,
      author = {Marklof, Jens and Strömbergsson, A},
      TITLE={ The periodic Lorentz gas in the Boltzmann-Grad limit: Asymptotic estimates},
      ARXIV={1008.3811},
      }
  • [visible] J. Marklof and A. Strömbergsson, The distribution of visible lattice points.
    @misc{visible,
      author = {Marklof, Jens and Strömbergsson, A},
      TITLE={The distribution of visible lattice points},
      NOTE={in preparation},
      }
  • [Melbourne05] Go to document I. Melbourne and M. Nicol, "Almost sure invariance principle for nonuniformly hyperbolic systems," Comm. Math. Phys., vol. 260, iss. 1, pp. 131-146, 2005.
    @article {Melbourne05, MRKEY = {2175992},
      AUTHOR = {Melbourne, Ian and Nicol, Matthew},
      TITLE = {Almost sure invariance principle for nonuniformly hyperbolic systems},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {260},
      YEAR = {2005},
      NUMBER = {1},
      PAGES = {131--146},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {37D25 (37A99 37C40 82B05)},
      MRNUMBER = {2006h:37047},
      MRREVIEWER = {Slobodan N. Simi{ć}},
      DOI = {10.1007/s00220-005-1407-5},
      ZBLNUMBER = {1084.37024},
      }
  • [Melbourne07] Go to document I. Melbourne and M. Nicol, "A vector-valued almost sure invariance principle for hyperbolic dynamical systems," Ann. Probab., vol. 37, iss. 2, pp. 478-505, 2009.
    @article {Melbourne07, MRKEY = {2510014},
      AUTHOR = {Melbourne, Ian and Nicol, Matthew},
      TITLE = {A vector-valued almost sure invariance principle for hyperbolic dynamical systems},
      JOURNAL = {Ann. Probab.},
      FJOURNAL = {The Annals of Probability},
      VOLUME = {37},
      YEAR = {2009},
      NUMBER = {2},
      PAGES = {478--505},
      ISSN = {0091-1798},
      CODEN = {APBYAE},
      MRCLASS = {37A50 (37D20 37D25 37D50 60F17 82C40)},
      MRNUMBER = {2010i:37016},
      MRREVIEWER = {Stefano Galatolo},
      DOI = {10.1214/08-AOP410},
      ZBLNUMBER = {1176.37006},
      }
  • [Polya18] G. Pólya, "Zahlentheoretisches und Wahrscheinlichkeitstheoretisches über die Sichtweite im Walde," Arch. Math. Phys., vol. 27, pp. 135-142, 1918.
    @article{Polya18,
      author={Pólya, G.},
      TITLE={Zahlentheoretisches und Wahrscheinlichkeitstheoretisches über die Sichtweite im Walde},
      JOURNAL={Arch. Math. Phys.},
      VOLUME={27},
      YEAR={1918},
      PAGES={135--142},
      }
  • [Ratner91] Go to document M. Ratner, "On Raghunathan’s measure conjecture," Ann. of Math., vol. 134, iss. 3, pp. 545-607, 1991.
    @article {Ratner91, MRKEY = {1135878},
      AUTHOR = {Ratner, Marina},
      TITLE = {On {R}aghunathan's measure conjecture},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {134},
      YEAR = {1991},
      NUMBER = {3},
      PAGES = {545--607},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {22E40 (58F11 58F17)},
      MRNUMBER = {93a:22009},
      MRREVIEWER = {S. G. Dani},
      DOI = {10.2307/2944357},
      ZBLNUMBER = {0763.28012},
      }
  • [Szasz07] Go to document D. Szász and T. Varjú, "Limit laws and recurrence for the planar Lorentz process with infinite horizon," J. Stat. Phys., vol. 129, iss. 1, pp. 59-80, 2007.
    @article {Szasz07, MRKEY = {2349520},
      AUTHOR = {Sz{á}sz, Domokos and Varj{ú},
      Tam{á}s},
      TITLE = {Limit laws and recurrence for the planar {L}orentz process with infinite horizon},
      JOURNAL = {J. Stat. Phys.},
      FJOURNAL = {Journal of Statistical Physics},
      VOLUME = {129},
      YEAR = {2007},
      NUMBER = {1},
      PAGES = {59--80},
      ISSN = {0022-4715},
      MRCLASS = {37D50 (37N20 60F05 82C40)},
      MRNUMBER = {2008k:37087},
      MRREVIEWER = {Nikolai Chernov},
      DOI = {10.1007/s10955-007-9367-0},
      ZBLNUMBER = {1128.82011},
      }
  • [Shah96] Go to document N. A. Shah, "Limit distributions of expanding translates of certain orbits on homogeneous spaces," Proc. Indian Acad. Sci. Math. Sci., vol. 106, iss. 2, pp. 105-125, 1996.
    @article {Shah96, MRKEY = {1403756},
      AUTHOR = {Shah, Nimish A.},
      TITLE = {Limit distributions of expanding translates of certain orbits on homogeneous spaces},
      JOURNAL = {Proc. Indian Acad. Sci. Math. Sci.},
      FJOURNAL = {Indian Academy of Sciences. Proceedings. Mathematical Sciences},
      VOLUME = {106},
      YEAR = {1996},
      NUMBER = {2},
      PAGES = {105--125},
      ISSN = {0253-4142},
      MRCLASS = {22E40},
      MRNUMBER = {98b:22024},
      MRREVIEWER = {Dave Witte Morris},
      DOI = {10.1007/BF02837164},
      ZBLNUMBER = {0864.22004},
      }
  • [Shimura] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Publications of the Mathematical Society of Japan, No. 11, Iwanami Shoten, Publishers, Tokyo, 1971.
    @book {Shimura, MRKEY = {0314766},
      AUTHOR = {Shimura, Goro},
      TITLE = {Introduction to the Arithmetic Theory of Automorphic Functions},
      PUBLISHER = {Publications of the Mathematical Society of Japan, No. 11, Iwanami Shoten, Publishers, Tokyo},
      YEAR = {1971},
      PAGES = {xiv+267},
      MRCLASS = {10D10 (12-02 14K22)},
      MRNUMBER = {47 \#3318},
      MRREVIEWER = {A. N. Andrianov},
      ZBLNUMBER = {0221.10029},
      }
  • [Shiryaev] A. N. Shiryaev, Probability, Second ed., New York: Springer-Verlag, 1996, vol. 95.
    @book {Shiryaev, MRKEY = {1368405},
      AUTHOR = {Shiryaev, A. N.},
      TITLE = {Probability},
      SERIES = {Grad. Texts in Math.},
      VOLUME = {95},
      EDITION = {Second},
      NOTE = {Translated from the first (1980) Russian edition by R. P. Boas},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1996},
      PAGES = {xvi+623},
      ISBN = {0-387-94549-0},
      MRCLASS = {60-01},
      MRNUMBER = {97c:60003},
      ZBLNUMBER = {0870.00038},
      }
  • [siegel] C. L. Siegel, Lectures on the Geometry of Numbers, New York: Springer-Verlag, 1989.
    @book {siegel, MRKEY = {1020761},
      AUTHOR = {Siegel, Carl Ludwig},
      TITLE = {Lectures on the Geometry of Numbers},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1989},
      PAGES = {x+160},
      ISBN = {3-540-50629-2},
      MRCLASS = {11Hxx (01A75)},
      MRNUMBER = {91d:11070},
      MRREVIEWER = {Rudolf Scharlau},
      ZBLNUMBER = {0691.10021},
      }
  • [Spohn78] Go to document H. Spohn, "The Lorentz process converges to a random flight process," Comm. Math. Phys., vol. 60, iss. 3, pp. 277-290, 1978.
    @article {Spohn78, MRKEY = {0496299},
      AUTHOR = {Spohn, Herbert},
      TITLE = {The {L}orentz process converges to a random flight process},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {60},
      YEAR = {1978},
      NUMBER = {3},
      PAGES = {277--290},
      ISSN = {0010-3616},
      MRCLASS = {82.60},
      MRNUMBER = {58 \#14861},
      MRREVIEWER = {Burton H. Voorhees},
      URL = {http://projecteuclid.org/getRecord?id=euclid.cmp/1103904131},
      ZBLNUMBER = {0381.60099},
      }
  • [SV] Go to document A. Strömbergsson and A. Venkatesh, "Small solutions to linear congruences and Hecke equidistribution," Acta Arith., vol. 118, iss. 1, pp. 41-78, 2005.
    @article {SV, MRKEY = {2136220},
      AUTHOR = {Str{ö}mbergsson, Andreas and Venkatesh, Akshay},
      TITLE = {Small solutions to linear congruences and {H}ecke equidistribution},
      JOURNAL = {Acta Arith.},
      FJOURNAL = {Acta Arithmetica},
      VOLUME = {118},
      YEAR = {2005},
      NUMBER = {1},
      PAGES = {41--78},
      ISSN = {0065-1036},
      CODEN = {AARIA9},
      MRCLASS = {11P21 (11F72)},
      MRNUMBER = {2006g:11204},
      MRREVIEWER = {A. Raghuram},
      DOI = {10.4064/aa118-1-4},
      ZBLNUMBER = {1082.11061},
      }

Authors

Jens Marklof

School of Mathematics
University of Bristol
Bristol, BS8 1TW
United Kingdom

Andreas Strömbergsson

Department of Mathematics
Box 480
Uppsala University
SE-75106 Uppsala
Sweden