Hardy-Szegő点过程：大偏差与强Szegő渐近
Hardy-Szegő Point Processes: Large Deviations and Strong Szegő Asymptotics
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中文总结 AI 辅助
该研究探究Hardy-Szegő零点过程的指数尺度涨落，确定极限标度对数矩生成函数，证明Nₐ(L)/L的大偏差原理与对数矩生成函数的强Szegő展开，为相关概率几何研究提供关键结果。
中文摘要 AI 辅助
我们研究由Peres和Virág在圆盘框架下研究的Hardy-Szegő零点过程的指数尺度涨落，该过程在上半平面的实现形式展现出不同的概率几何结构。这个共形不变的行列式零点过程与圆盘实现等价，但上半平面坐标使实平移不变性明确显现，并将长水平窗口选为自然观测对象。对于从高度1到高度a>1的垂直窗口，记Nₐ(L)为对应长度L的水平窗口内的零点数量，我们明确确定了极限标度对数矩生成函数。由此，我们证明了Nₐ(L)/L的大偏差原理，其速率函数由该极限的勒让德变换给出。我们还证明了对数矩生成函数的强Szegő展开，包括显式的一阶修正项，该修正项在自然复带内局部一致成立。证明采用投影前的平面行列式结构，将问题简化为一维Fredholm行列式，并结合固定幂次迹渐近与Wiener-Hopf比较方法。
英文摘要
We study exponential-scale fluctuations of the Hardy-Szegő zero process, investigated by Peres and Virág in the disk setting, in its upper-half-plane realization, which reveals a different probabilistic geometry. This conformally invariant determinantal zero process is equivalent to its disk realization, but the upper-half-plane coordinates make real-translation invariance explicit and single out long horizontal windows as natural observables. For the vertical window from height one to height $a>1$, let $N_a(L)$ denote the number of zeros in the corresponding horizontal window of length $L$. We identify the limiting scaled log-moment generating function explicitly. As a consequence, we prove a large deviations principle for $N_a(L)/L$, with rate function given by the Legendre transform of this limit. We also prove a strong Szegő expansion for the log-moment generating function, including an explicit order-one correction, locally uniformly in the natural complex strip. The proof uses the planar determinantal structure before projection, reduces the problem to a one-dimensional Fredholm determinant, and combines fixed-power trace asymptotics with a Wiener-Hopf comparison.
