论文标题
随机矩阵动力学的liouville量子重力
Liouville quantum gravity from random matrix dynamics
论文作者
论文摘要
我们建立了$ 2D $ liouville量子重力与随机矩阵的自然动态之间的第一个连接。特别是，我们表明，如果$（u_t）$是以平衡为单位的布朗人动议，那么衡量$$ | \ det（u_t -e^{iθ}）|^γdtdt除timimential的$ 2D $ lqg量度，$ 2D $ lqg衡量$ 2D $ 2D $ 2D $ 2D $ 2D。自韦伯（Webb，2014年的工作）以来，提出了与这些动力学相关的高斯自由场类型波动，并首先建立了2D $设置的LQG度量，以$ 2D $设置的融合，他证明了通过使用Riemann-Hilbert理论的投入来证明相关的一项维度措施的收敛。 收敛性来自Widom（1973）对带有真实符号的Toeplitz决定因素的Fisher-Hartwig渐近学的第一个多时间扩展。为了证明这些证明，我们提出了一项一般手术论点，并将确定点过程估计与对谎言组的随机分析相结合，提供了韦伯$ 1D $结果的概率证明。我们认为，这些技术将更广泛地适用于矩阵动力学，从平衡，相关随机矩阵类别的决定因素的联合力矩以及非弱者随机矩阵的特征多项式。
We establish the first connection between $2d$ Liouville quantum gravity and natural dynamics of random matrices. In particular, we show that if $(U_t)$ is a Brownian motion on the unitary group at equilibrium, then the measures $$ |\det(U_t - e^{i θ})|^γ dt dθ$$ converge in the limit of large dimension to the $2d$ LQG measure, a properly normalized exponential of the $2d$ 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 $2d$ 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 (1973) 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 $1d$ 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.
