M.Sc Student | Zuck Reut |
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Subject | The limiting distribution of random Cayley Graphs |

Department | Department of Mathematics |

Supervisor | Professor Uri Shapira |

Full Thesis text |

We study the distribution of diameters of Cayley graphs of Z^{n}/Sigma,
where Sigma is a full rank sublattice of Z^{n}, these graph are also
known as multidimensional circulant graphs. In our study we consider random
Cayley graphs of Z^{n} /Sigma and we take the generating set of the
graph to be of fixed size and to be chosen uniformly at random such that the
resulting graph is connected. We establish a limit distribution theorem for the
diameters of these random graphs. Our proof is based on connecting the diameter
of the Cayley graph with the covering radius of a sublattice of Z^{m}
of covolume 1, where m is the size of the generating set of the graph. We then
use an equidistribution result regarding the distribution of Hecke points, and
apply this theorem to the lattices which corresponds to the Cayley graphs. This
is a generalization of the work that was done by J. Marklof and A.
Strömbergsson where they used the limit distribution of Frobenius numbers
on m variables to study the distributions of the diameters of random Cayley
graphs of Z/kZ.