# Yuhao Xue's research while affiliated with Tsinghua University and other places

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## Publications (11)

In this paper, we investigate the asymptotic behavior of the non-simple systole, which is the length of a shortest non-simple closed geodesic, on a random closed hyperbolic surface on the moduli space $\mathcal{M}_g$ of Riemann surfaces of genus $g$ endowed with the Weil-Petersson measure. We show that as the genus $g$ goes to infinity, the non-sim...

In this paper, we investigate basic geometric quantities of a random hyperbolic surface of genus with respect to the Weil–Petersson measure on the moduli space . We show that as goes to infinity, a generic surface satisfies asymptotically: the separating systole of is approximately
there is a half‐collar of width approximately around any separating...

Let $\mathcal{M}_g$ be the moduli space of hyperbolic surfaces of genus $g$ endowed with the Weil-Petersson metric. In this paper, we show that for any $\epsilon>0$, as $g\to \infty$, for a generic surface in $\mathcal{M}_g$, the error term in the Prime Geodesic Theorem is bounded from above by $g\cdot t^{\frac{3}{4}+\epsilon}$, up to a uniform con...

Let Mg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {M}_g$$\end{document} be the moduli space of hyperbolic surfaces of genus g endowed with the Weil–Peterss...

In this note we show that the expected value of the separating systole of a random surface of genus g with respect to Weil–Petersson volume behaves like $2\log g $ as the genus goes to infinity. This is in strong contrast to the behavior of the expected value of the systole which, by results of Mirzakhani and Petri, is independent of genus.

In this article we study the first eigenvalues of closed Riemann surfaces for large genus. We show that for every closed Riemann surface $X_g$ of genus $g$ $(g\geq 2)$, the first eigenvalue of $X_g$ is greater than $\frac{\mathcal{L}_1(X_g)}{g^2}$ up to a uniform positive constant multiplication. Where $\mathcal{L}_1(X_g)$ is the shortest length of...

Let $M_g$ be the moduli space of hyperbolic surfaces of genus $g$ endowed with the Weil-Petersson metric. In this paper, we show that for any $\epsilon>0$, as genus $g$ goes to infinity, a generic surface $X\in M_g$ satisfies that the first eigenvalue $\lambda_1(X)>\frac{3}{16}-\epsilon$.

In this note we show that the expected value of the separating systole of a random surface of genus $g$ with respect to Weil-Petersson volume behaves like $2\log g $ as the genus goes to infinity.

In this paper, we investigate basic geometric quantities of a random hyperbolic surface of genus $g$ with respect to the Weil-Petersson measure on the moduli space $\mathcal{M}_g$. We show that as $g$ goes to infinity, a generic surface $X\in \mathcal{M}_g$ satisfies asymptotically: (1) the separating systole of $X$ is about $2\log g$; (2) there is...

In this article we study the asymptotic behavior of small eigenvalues of hyperbolic surfaces for large genus. We show that for any positive integer k k , as the genus g g goes to infinity, the minimum of k k -th eigenvalues of hyperbolic surfaces over any thick part of moduli space of Riemann surfaces of genus g g is uniformly comparable to 1 g 2 \...

## Citations

... Remark. It was shown in [NWX23,Theorem 4] that for any ϵ > 0, lim g→∞ Prob g WP X ∈ M g ; (1 − ϵ) log g < ℓ ns sys (X) < 2 log g = 1. ...

... For others asymptotic properties see e. g. Lipnowski and Wright [15] and Wu and Xue [24]. ...

... The geometry and spectra of random hyperbolic surfaces under this Weil-Petersson measure have been widely studied in recent years. For examples, one may see [GPY11] for Bers' constant, [Mir13,WX22b] for diameter, [MP19] for systole, [Mir13,NWX23,PWX22] for separating systole, [Mir13,WX22b,LW21,AM23] for first eigenvalue, [GMST21] for eigenfunction, [Mon22] for Weyl law, [Rud22,RW23] for GOE, [WX22a] for prime geodesic theorem, [Nau23] for determinant of Laplacian. One may also see [LS20, MT21, Hid22, SW22, HW22, HHH22, HT22, DS23, Gon23,MS23] and the references therein for more related topics. ...

... (2) Based on [NWX23], joint with Parlier, the third and forth named authors in [PWX22] showed that lim g→∞ Mg ℓ sep sys (X)dX Vol WP (M g ) log g = 2 ...

... By elementary Isoperimetric Inequality (e.g. see [Bus92,WX21]) we know that ...