Search results for "60G70"
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From $1$ to $6$: a finer analysis of perturbed branching Brownian motion
2020
The logarithmic correction for the order of the maximum for two-speed branching Brownian motion changes discontinuously when approaching slopes $\sigma_1^2=\sigma_2^2=1$ which corresponds to standard branching Brownian motion. In this article we study this transition more closely by choosing $\sigma_1^2=1\pm t^{-\alpha}$ and $\sigma_2^2=1\pm t^{-\alpha}$. We show that the logarithmic correction for the order of the maximum now smoothly interpolates between the correction in the iid case $\frac{1}{2\sqrt 2}\ln(t),\;\frac{3}{2\sqrt 2}\ln(t)$ and $\frac{6}{2\sqrt 2}\ln(t)$ when $0<\alpha<\frac{1}{2}$. This is due to the localisation of extremal particles at the time of speed change which depen…
Maxima of a Random Model of the Riemann Zeta Function over Intervals of Varying Length
2021
We consider a model of the Riemann zeta function on the critical axis and study its maximum over intervals of length $(\log T)^{\theta}$, where $\theta$ is either fixed or tends to zero at a suitable rate. It is shown that the deterministic level of the maximum interpolates smoothly between the ones of log-correlated variables and of i.i.d. random variables, exhibiting a smooth transition 'from $\frac34$ to $\frac14$' in the second order. This provides a natural context where extreme value statistics of log-correlated variables with time-dependent variance and rate occur. A key ingredient of the proof is a precise upper tail tightness estimate for the maximum of the model on intervals of si…