We study random Bernoulli measures $\mu_{\mathbf{p}}$ in a random environment $\mathbf{p}=(p_1,p_2,\ldots)$, where $p_n$ are independent and identically distributed on $(0,1)$. For almost every environment $\mathbf{p}$, we establish several almost sure properties of $\mu_{\mathbf{p}}$: its local dimension, $L^q$ dimensions, Rajchman property, mutual singularity with any fixed measure, and a dichotomy for normal numbers of its typical points.
Let $M_n$ be an $n\times n$ matrix with independent uniform sign entries. We prove that there exist absolute constants $C,c>0$ such that, for all sufficiently large $n$, \[ \mathbb{P}\!\left( \left|\operatorname{Per}(M_n)\right| \ge e^{-Cn}\sqrt{n!} \right) \ge 1-n^{-c}. \] This establishes the exponential scale lower...
We discuss the famous conjecture {$\theta(p_c,{\mathbb Z}^d)=0$} for the Bernoulli site percolation model on $\Z^d$ with $d\geq 3$. We prove that, for $d\geq 3$, in the Bernoulli percolation model on ${\mathbb Z}^d$, at the critical point $p_c$, either there is no infinite cluster or the tail of the finite cluster dist...
This paper investigates the discrepancy of a family of random sampling methods obtained by perturbing the grid $\frac{1}{M}\mathbb{Z}^{d}\cap\left[ -1/2,1/2\right)^{d}$, where $M$ is a large positive integer. Parameterized by an arbitrary probability measure $\mu$, this family encompasses several classical methods for...
R. Bramati, L. Brandolini, G. Travaglini· 0 citations
We consider a random walk $\{S_n\}$ with a finite positive drift that is stopped at a random time $\tau$ having an intermediate regularly varying distribution. We assume that the jump distribution is lighter-tailed than the distribution of $\tau$. Under these conditions, we show that the tails of the distributions of $...
We prove that the path $P_{11}$ does not admit uniform average mixing under any probability distribution on $\mathbb R$, answering a question of Baptista, Coutinho, and Marques in the negative. The proof is exact: we construct an explicit rational symmetric matrix $Y$ such that $\langle Y,M(t)\rangle_F=1$ for every $t\...
We prove that, for either a Steinhaus random multiplicative function or a Rademacher random multiplicative function $f$, and every $\varepsilon>0$, almost surely $|\sum_{n\le x}f(n)|\ll_{\varepsilon,f}\sqrt{x}(\log\log x)^{1/4+\varepsilon}$. Together with Harper's almost sure lower bound, this determines the sharp loga...
Benjamin Durkan, Andrew Pearce-Crump· 2 citations· ⚡1
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