Matilde Lalin : The distribution of points on cyclic l-covers of genus g
- Number Theory ( 162 Views )We give an overview of a general trend of results that say that the distribution of the number of F_q-points of certain families of curves of genus g is asymptotically given by a sum of q+1 independent, identically distributed random variables as g goes to infinity. In particular, we discuss the distribution of the number of F_q-points for cyclic l-covers of genus g. (This is joint work with Bucur, David, Feigon, Kaplan, Ozman, Wood.) This work generalizes previous results in which only connected components of the moduli space were considered.
Rick Durrett : Diffusion limit for the partner model at the critical value
- Probability ( 104 Views )The partner model is an SIS epidemic in a population with random formation and dissolution of partnerships, and disease transmission only occurs within partnerships. Foxall, Edwards, and van den Driessche found the critical value and studied the subcritical and supercritical regimes. Recently Foxall has shown that (if there are enough initial infecteds) then the critical model survives for time \(O(N^{1/2})\). Here we improve that result by proving the convergence of \(i_N(t)=I(tN^{1/2})/N^{1/2}\) to a limiting diffusion. We do this by showing that in the first O(1), this four dimensional process collapses to two dimensions: the number of SI and II partnerships are constant multiples of the the number of infected singles \(I_t\). The other variable \(Y_t\), the total number of singles, behaves like an Ornstein-Uhlenbeck process on a time scale O(1) and averages out of the limit theorem for \(i_N(t)\). This is joint work with Anirban Basak and Eric Foxall.