Tightness of the recentered maximum of log-correlated Gaussian fields

Tightness of the recentered maximum of the two-dimensional discrete Gaussian Free Field

We consider the maximum of the discrete two dimensional Gaussian free field (GFF) in a box, and prove that its maximum, centered at its mean, is tight, settling a long-standing conjecture. The proof combines a recent observation of Bolthausen, Deuschel and Zeitouni with elements from (Bramson 1978) and comparison theorems for Gaussian fields. An essential part of the argument is the precise evaluation, up to an error of order 1, of the expected value of the maximum of the GFF in a box. Related Gaussian fields, such as the GFF on a two-dimensional torus, are also discussed.

Discretization of the maximum for the derivatives of a random model of $\log|ζ|$ on the critical line

In this short note, we study the derivatives of all orders for the random field $$ X_T(h) = \sum_{p \leq T} \frac{\text{Re}(U_p \, p^{-i h})}{p^{1/2}}, \quad h\in [0,1], $$ where $(U_p, \, p ~\text{primes})$ is an i.i.d. sequence of uniform random variables on the unit circle in $\mathbb{C}$. We show that the maximum of $X_{T}$, and more generally the maximum of its $j$-th derivative, varies on a $(\log T)^{-\frac{1}{2}(j+2)}$ scale, which improves and extends the main result in Arguin & Ouimet (2019) and makes further progress towards the open problem of the tightness of the recentered maximu…

微观减速高斯场的极值

Extrema of cooling branching Brownian motion and related Gaussian fields

引入一族协方差结构具有非均匀微观减速的高斯场，研究其最大值的渐近行为，发现一阶增长指数随参数α线性变化，而二阶修正项在α=1/3处发生相变。

任意维度下对数相关高斯场指数度量的紧致性

Tightness of exponential metrics for log-correlated Gaussian fields in arbitrary dimension

研究\(\mathbb{R}^d\)（\(d\geq2\)）上类似刘维尔量子引力度量的自然近似方案，通过特定高斯随机函数序列构建随机度量族，证明在亚临界阶段经重标度后度量紧致，子序列极限诱导欧几里得拓扑，并列出诸多开放问题。

Large deviations and continuity estimates for the derivative of a random model of $\log |ζ|$ on the critical line

In this paper, we study the random field \begin{equation*} X(h) \circeq \sum_{p \leq T} \frac{\text{Re}(U_p \, p^{-i h})}{p^{1/2}}, \quad h\in [0,1], \end{equation*} where $(U_p, \, p ~\text{primes})$ is an i.i.d. sequence of uniform random variables on the unit circle in $\mathbb{C}$. Harper (2013) showed that $(X(h), \, h\in (0,1))$ is a good model for the large values of $(\log |ζ(\frac{1}{2} + i (T + h))|, \, h\in [0,1])$ when $T$ is large, if we assume the Riemann hypothesis. The asymptotics of the maximum were found in Arguin, Belius & Harper (2017) up to the second order, but the tightn…

Mixture-based Modeling of Spatially Correlated Interference in a Poisson Field of Interferers

As the interference in PPP based wireless networks exhibit spatial correlation, any joint analysis involving multiple spatial points either end up with numerical integrations over $\mathbb{R}^2$ or become analytically too intractable. To tackle these issues, we present an alternate approach which not only offers a simpler analytical structure, but also closely mimics the PPP characteristics. This approach at its core models the correlated interferences using a correlation framework constructed using random variable mixtures. Additionally, a correlation framework based on the more standard meth…