Mikhail Lifshits, Michel Weber
We study the supremum of some random Dirichlet polynomials with independent coefficients and obtain sharp upper and lower bounds for supremum expectation thus extending the results from our previous work (see http://arXiv.org/abs/math/0703691). Our approach in proving these results is entirely based on methods of stochastic processes, in particular the metric entropy method.
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We study the supremum of some random Dirichlet polynomials and obtain sharp upper and lower bounds for supremum expectation that extend the optimal estimate of Halász-Queffélec and enable to cunstruct random polynomials with unusually small maxima. Our approach in proving these results is entirely based on methods of stochastic processes, in particular the metric entropy method.
We study the supremum of random Dirichlet polynomials $D_N(t)=\sum_{n=1}^N\varepsilon_n d(n) n^{- s}$, where $(\varepsilon_n)$ is a sequence of independent Rademacher random variables, and $ d $ is a sub-multiplicative function. The approach is gaussian and entirely based on comparison properties of Gaussian processes, with no use of the metric entropy method.
In the first part of this expository paper, we present and discuss the interplay of Dirichlet polynomials in some classical problems of number theory, notably the Lindelöf Hypothesis. We review some typical properties of their means and continue with some investigations concerning their supremum properties. Their random counterpart is considered in the last part of the paper, where a analysis of their supremum properties, based on methods of stochastic processes, is developed.
Joint density of a stable process and its supremum: regularity and upper bounds
本文利用模拟中的三个思想，结合凹极大子表示和Chambers-Mallows-Stuck表示，通过多层蒙特卡罗插值加速，建立了稳定过程及其上确界联合密度的无限可微性和近乎最优多项式上界。
We give a polynomial time approximation scheme (PTAS) for computing the supremum of a Gaussian process. That is, given a finite set of vectors $V\subseteq\mathbb{R}^d$, we compute a $(1+\varepsilon)$-factor approximation to $\mathop {\mathbb{E}}_{X\leftarrow\mathcal{N}^d}[\sup_{v\in V}|\langle v,X\rangle|]$ deterministically in time $\operatorname {poly}(d)\cdot|V|^{O_{\varepsilon}(1)}$. Previously, only a constant factor deterministic polynomial time approximation algorithm was known due to the work of Ding, Lee and Peres [Ann. of Math. (2) 175 (2012) 1409-1471]. This answers an open question…
A Polynomial Rate of Convergence for the Dirichlet Problem on Orthodiagonal Maps
本文推广了先前工作，证明了正交对角映射上带Hölder边界数据的Dirichlet问题具有多项式收敛速率，关键方法是利用光滑磨光子使离散调和函数近似调和，并由此得到介观尺度上的Lipschitz性质。
Comparison of Dirichlet forms for stable-like random walks on groups of polynomial volume growth
本研究针对多项式体积增长群上由特定测度驱动的类稳定随机游动，建立了其关联狄利克雷形式等价性的几何与代数判定准则，用于比较两类随机游动的行为。
For $X(n)$ a Steinhaus random multiplicative function, we study the maximal size of the random Dirichlet polynomial $$ D_N(t) = \frac1{\sqrt{N}} \sum_{n \leq N} X(n) n^{it}, $$ with $t$ in various ranges. In particular, for fixed $C>0$ and any small $\varepsilon>0$ we show that, with high probability, $$ \exp( (\log N)^{1/2-\varepsilon} ) \ll \sup_{|t| \leq N^C} |D_N(t)| \ll \exp( (\log N)^{1/2+\varepsilon}). $$
We study extreme values of Dirichlet polynomials with multiplicative coefficients, namely \[D_N(t) : = D_{f,\, N}(t)= \frac{1}{\sqrt{N}} \sum_{n\leqslant N} f(n) n^{it}, \] where $f$ is a completely multiplicative function with $|f(n)|=1$ for all $n\in\mathbb{N}$. We use Soundararajan's resonance method to produce large values of $\left|D_N(t)\right|$ uniformly for all such $f$. In particular, we improve a recent result of Benatar and Nishry, where they establish weaker lower bounds and only for almost all such $f$.
Dirichlet law for factorization of integers, polynomials and permutations
本文通过多维围道积分证明整数、多项式和排列分解为 $k$ 部分时遵循狄利克雷分布，推广了 DDT 反正弦定律，并指出任意参数的狄利克雷分布可类似建模。
On the expected number of roots of a random Dirichlet polynomial
本文研究随机狄利克雷多项式在二进区间内根的期望个数，证明其近似为 $2/\sqrt{3}$ 倍黎曼 $ζ$ 函数零点数，并推广至导数情形。
We look at random walks in Dirichlet environment. It was known that in dimension $d\geq 3$, if the walk is sub-ballistic, the displacement of the walk is polynomial of order $κ$ for some explicit $κ$. We show that the walk, after renormalization, actually converges to a $κ$-stable completely asymmetric Levy Process.
We consider a multivariate version of the so-called Lancaster problem of characterizing canonical correlation coefficients of symmetric bivariate distributions with identical marginals and orthogonal polynomial expansions. The marginal distributions examined in this paper are the Dirichlet and the Dirichlet multinomial distribution, respectively, on the continuous and the N-discrete d-dimensional simplex. Their infinite-dimensional limit distributions, respectively, the Poisson-Dirichlet distribution and Ewens's sampling formula, are considered as well. We study, in particular, the possibility…
We consider random walks in Dirichlet environment (RWDE) on $\Z ^d$, for $ d \geq 3 $, in the sub-ballistic case. We associate to any parameter $ (α_1, ..., α_{2d}) $ of the Dirichlet law a time-change to accelerate the walk. We prove that the continuous-time accelerated walk has an absolutely continuous invariant probability measure for the environment viewed from the particle. This allows to characterize directional transience for the initial RWDE. It solves as a corollary the problem of Kalikow's 0-1 law in the Dirichlet case in any dimension. Furthermore, we find the polynomial order of th…
The spectral measures of random Jacobi matrices related to beta ensembles at high temperature and Dirichlet processes
本文研究了高温下β集合随机雅可比矩阵的谱测度，证明其收敛于狄利克雷过程并提供显式谱测度的例子。
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