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Tianhao Xian

Publications and source records attributed to Tianhao Xian.

5 recordsLinked to original sources
arXiv2025

Edge homogenization of Dyson Brownian motion and applications

We prove a homogenization result for the difference of two coupled Dyson Brownian motions started from generalized Wigner matrix initial data. We prove an optimal order, high probability estimate that is valid throughout the spectrum, including up to the spectral edges. Prior homogenization results concerned only the bulk of the spectrum. We apply our estimate to address the question of quantifying edge universality. Here, we have two results. We show that the Kolmogorov-Smirnov distance of the distribution of the gap between the largest two eigenvalues of a generalized Wigner matrix (with smooth entry distribution) and its GOE/GUE counterpart is $\mathcal{O}(N^{-1+\varepsilon})$. On the other hand, we show that, for the distribution of the largest eigenvalue, there are Wigner matrices so that the analogous Kolmogorov-Smirnov distance is bounded below by $N^{-1/3-\varepsilon}$.

math.PR↗
arXiv2022

Spontaneous synchronisation and exceptional points in breather complex

We experimentally demonstrate the spontaneous synchronization and the exceptional point (EP) induced pulse generation mechanism in the breather complex. The breathing frequency and phase are found to be synchronized during the formation of a 9-breather assembled complex in a mode-locked fiber laser. The breathers are formed at exactly the time point of the complex's breathing frequency leaving or entering the subharmonic entrainment. Such new pulse generation mechanism should be related to the non-Hermitian EPs or the time translation symmetry breaking. The investigations of destroying and rebuilding the mode-locking reveal the connection between the synchronization and laser stabilization. These findings may inspire a wide range of researches including ultrafast optics, micro-cavity combs, ocean breather behaviors, non-Hermitian optics, etc.

physics.optics↗
arXiv2021

Dispersive temporal holography for single-shot recovering comprehensive ultrafast dynamics

It is critical to characterize the carrier and instantaneous frequency distribution variation in ultrafast processes, all of which are determined by the optical phase. Nevertheless, there is no method that can single-shot record the intro-pulse phase evolution of pico/femtosecond signals, to date. By analogying holographic principle in space to the time domain and using the time-stretch method, we propose the dispersive temporal holography to single-shot recover the phase and amplitude of ultrafast signals. It is a general and comprehensive technology and can be applied to analyze ultrafast signals with highly complex dynamics. The method provides a new powerful tool for exploring ultrafast science, which may benefit many fields, including laser dynamics, ultrafast diagnostics, nonlinear optics, and so on.

physics.optics↗
arXiv2021

Quasi-invariance of fractional Gaussian fields nonlinear wave equation with polynomial nonlinearity

We prove quasi-invariance of Gaussian measures $μ_s$ with Cameron-Martin space $H^s$ under the flow of the defocusing nonlinear wave equation with polynomial nonlinearities of any order for all $s>5/2$, including fractional $s$. This extends work of Oh-Tzvetkov [15] and Gunaratnam-Oh-Tzvetkov-Weber [10] who proved this result this result for a cubic nonlinearity and $s$ an even integer. The main contributions are a modified construction of a weighted adapted to higher order nonlinearity, and an energy estimate for the derivative of the energy replacing the integration by parts argument introduced in previous works. We also address the question of (non) quasi-invariance for the dispersionless model raised in the introductions to [15,10].

math.AP↗