On the Poisson distribution of lengths of lattice vectors in a random lattice

TY - JOUR

T1 - On the Poisson distribution of lengths of lattice vectors in a random lattice

AU - Södergren, Carl Anders

PY - 2011/12

Y1 - 2011/12

N2 - We prove that the volumes determined by the lengths of the non-zero vectors ±x in a random lattice L of covolume 1 define a stochastic process that, as the dimension n tends to infinity, converges weakly to a Poisson process on the positive real line with intensity 1/2. This generalizes earlier results by Rogers (Proc Lond Math Soc (3) 6:305-320, 1956, Thm. 3) and Schmidt (Acta Math 102:159-224, 1959, Satz 10).

AB - We prove that the volumes determined by the lengths of the non-zero vectors ±x in a random lattice L of covolume 1 define a stochastic process that, as the dimension n tends to infinity, converges weakly to a Poisson process on the positive real line with intensity 1/2. This generalizes earlier results by Rogers (Proc Lond Math Soc (3) 6:305-320, 1956, Thm. 3) and Schmidt (Acta Math 102:159-224, 1959, Satz 10).

M3 - Journal article

SN - 0025-5874

VL - 269

SP - 945

EP - 954

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 3-4

ER -