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 confirms, up to the exponential scale, the lower bound suggested by Tao and Vu.
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 distribution is not a stretched exponential.
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 evaluating the quality of an $N$-point set in $\mathbb{T}^{d}$, where $N=M^{d}$. We show that all probability measures, except for Dirac measures, behave like the Lebesgue measure in the Monte Carlo discrepancy. This represents a limiting case where the measure $\mu$ depends on $M$. In this latter context, we prove that, up to a constant, the lowest possible discrepancy is achieved when the support of $\mu$ has diameter $\leq c/M$, and that this upper bound is sharp.
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 $S_{\tau}$ and $M_{\tau} = \max_{k\le \tau} S_k$ are asymptotically equivalent and are determined by the tail of $\tau$, while the random walk $\{S_n\}$ contributes only through the law of large numbers.
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 $t\in\mathbb R$, whereas $\langle Y,J/11\rangle_F=12/11$.
We prove that, for either a Steinhaus random multiplicative function or a Rademacher random multiplicative function $f$, and every $\eps>0$, almost surely $$ \left|\sum_{n\le x}f(n)\right|\ll_{\eps,f}\sqrt{x}(\log\log x)^{1/4+\eps}. $$ Together with Harper's almost sure lower bound, this determines the sharp logarithmic exponent in both models. This settles Harper's conjecture on the large fluctuations of random multiplicative functions.
Benjamin Durkan, Andrew Pearce‐Crump· 1 citation· ⚡1
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