标题： 关于随机尖点形式的上确界 显示英文标题

标题： On the supremum of random cusp forms

摘要： 引入了一个全模群的尖点形式的随机系综。 对于权为$k$的尖点形式，在模面的紧子域上，其期望最大值的真实量级被确定为$\asymp \sqrt{\log{k}}$，这与猜想的界一致。 此外，还建立了最大值在其中位数附近的指数集中性。 与紧情况相反，证明了全局期望最大值（在尖点附近取得）的增长率类似于$k^{1/4}$，最多相差一个对数因子。 显示英文摘要

摘要： A random ensemble of cusp forms for the full modular group is introduced. For a weight-$k$ cusp form, restricted to a compact subdomain of the modular surface, the true order of magnitude of its expected supremum is determined to be $\asymp \sqrt{\log{k}}$, in line with the conjectured bounds. Additionally, the exponential concentration of the supremum around its median is established. Contrary to the compact case, it is shown that the global expected supremum, which is attained around the cusp, grows like $k^{1/4}$, up to a logarithmic factor.