A Central Limit Theorem in Many-Body Quantum Dynamics

We study the many body quantum evolution of bosonic systems in the mean field limit. The dynamics is known to be well approximated by the Hartree equation. So far, the available results have the form of a law of large numbers. In this paper we go one step further and we show that the fluctuations around the Hartree evolution satisfy a central limit theorem. Interestingly, the variance of the limiting Gaussian distribution is determined by a time-dependent Bogoliubov transformation describing the dynamics of initial coherent states in a Fock space representation of the system.

Biased random walks on a Galton-Watson tree with leaves

We consider a biased random walk $X_n$ on a Galton-Watson tree with leaves in the sub-ballistic regime. We prove that there exists an explicit constant $gamma= gamma(beta) in (0,1)$, depending on the bias $beta$, such that $X_n$ is of order $n^{gamma}$. Denoting $Delta_n$ the hitting time of level $n$, we prove that $Delta_n/n^{1/gamma}$ is tight. Moreover we show that $Delta_n/n^{1/gamma}$ does not converge in law (at least for large values of $beta$). We prove that along the sequences $n_{lambda}(k)=lfloor lambda beta^{gamma k}rfloor$, $Delta_n/n^{1/gamma}$ converges to certain infinitely divisible laws. Key tools for the proof are the classical Harris decomposition for Galton-Watson trees, a new variant of regeneration times and the careful analysis of triangular arrays of i.i.d. heavy-tailed random variables.