Liouville quantum gravity from random matrix dynamics
Liouville quantum gravity from random matrix dynamics
复制标题
随机矩阵动力学中的刘维尔量子引力
DOI：
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发表时间：
2022
期刊：
影响因子：
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通讯作者：
Hugo Falconet
中科院分区：
文献类型：
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作者：
P. Bourgade;Hugo Falconet
我们建立了二维Liouville量子引力和随机矩阵的自然动力学之间的fi第一联系。特别地，我们证明了如果(Ut)是平衡点酉群上的布朗运动，则测度在大维极限上收敛到二维LQG测度，它是二维高斯自由fi场的适当归一化指数.Spohn(1998年)首次建立了与这些动力学有关的无高斯fi-eld型fl函数，并从fi(2014)的工作中猜想了在二维环境下收敛到LQG测度，他利用Riemann-Hilbert理论的输入证明了相关的一维测度的收敛。这种收敛源于Widom关于实符号Toeplitz行列式的Fisher-Hartwig渐近性的fi第一次多时间推广。为了证明这些，我们发展了一般的外科论证，并将行列式点过程估计与李群上的随机分析相结合，顺便给出了Webb的一维结果的概率证明。我们相信，这些技术将更广泛地应用于非平衡矩阵动力学、相关随机矩阵的行列式的联合矩以及非厄米特随机矩阵的特征多项式。
We establish the first connection between 2 d Liouville quantum gravity and natural dynamics of random matrices. In particular, we show that if ( U t ) is a Brownian motion on the unitary group at equilibrium, then the measures converge in the limit of large dimension to the 2 d LQG measure, a properly normalized exponential of the 2 d Gaussian free field. Gaussian free field type fluctuations associated with these dynamics were first established by Spohn (1998) and convergence to the LQG measure in 2 d settings was conjectured since the work of Webb (2014), who proved the convergence of related one dimensional measures by using inputs from Riemann-Hilbert theory. The convergence follows from the first multi-time extension of the result by Widom on Fisher-Hartwig asymptotics of Toeplitz determinants with real symbols. To prove these, we develop a general surgery argument and combine determinantal point processes estimates with stochastic analysis on Lie group, providing in passing a probabilistic proof of Webb’s 1 d result. We believe the techniques will be more broadly applicable to matrix dynamics out of equilibrium, joint moments of determinants for classes of correlated random matrices, and the characteristic polynomial of non-Hermitian random matrices.
RosenzweigâPorter 模型中的非遍历离域
影响因子：
1.2
作者：
P. von Soosten;S. Warzel
通讯作者：
S. Warzel
影响因子：
0.5
作者：
Berestycki, Nathanael
通讯作者：
Berestycki, Nathanael