Asymptotic distribution of hitting times for critical maps of the circle

It is well known that the renormalization group transformation $\mathcal{R}$ has a unique fixed point $f_{cr}$ in the space of critical $C^{3}$-circle homeomorphisms with one cubic critical point $x_{cr}$ and the golden mean rotation number $\overline{\rho}:=\frac{\sqrt{5}-1}{2}.$ Denote by $Cr(\overline{\rho})$ the set of all critical circle maps $C^{1}$-conjugated to $f_{cr}.$ Let $f\in Cr(\overline{\rho})$ and let $\mu:=\mu_{f}$ be the unique probability invariant measure of $f.$ Fix $\theta \in(0,1).$ For each $n\geq1$ define $c_{n}:=c_{n}(\theta)$ such that $\mu([x_{cr},c_{n}])=\theta\cdot\mu([x_{cr},f^{q_{n}}(x_{cr})]),$ where $q_{n}$ is the first return time of the linear rotation $f_{\overline{\rho}}.$ We study convergence in law of rescaled point process of time hitting. We show that the limit distribution is singular w.r.t. the Lebesgue measure.

Site Frequency Spectrum of the Bolthausen-Sznitman Coalescent

We derive explicit formulas for the two first moments of he site frequency spectrum (SFSn,b)1≤b≤n−1 of the Bolthausen-Sznitman coalescent along with some precise and efficient approximations, even for small sample sizes n. These results provide new L2-asymptotics for some values of b = o(n). We also study the length of internal branches carrying b > n/2 individuals. In this case we obtain the distribution function and a convergence in law. Our results rely on the random recursive tree construction of the Bolthausen-Sznitman coalescent.

Convergence of tandem Brownian queues

It is known that in a stationary Brownian queue with both arrival and service processes equal in law to Brownian motion, the departure process is a Brownian motion, identical in law to the arrival process: this is the analogue of Burke's theorem in this context. In this paper we prove convergence in law to this Brownian motion in a tandem network of Brownian queues: if we have an arbitrary continuous process, satisfying some mild conditions, as an initial arrival process and pass it through an infinite tandem network of queues, the resulting process weakly converges to a Brownian motion. We assume independent and exponential initial workloads for all queues.

Kloosterman paths and the shape of exponential sums

We consider the distribution of the polygonal paths joining partial sums of classical Kloosterman sums$\text{Kl}_{p}(a)$, as$a$varies over$\mathbf{F}_{p}^{\times }$and as$p$tends to infinity. Using independence of Kloosterman sheaves, we prove convergence in the sense of finite distributions to a specific random Fourier series. We also consider Birch sums, for which we can establish convergence in law in the space of continuous functions. We then derive some applications.