Harmonic Dirichlet Functions on Planar Graphs

大温度下通过狄利克雷过程的矩阵谐波分析

Matrix harmonic analysis at high temperature via the Dirichlet process

本文通过狄利克雷过程研究大温度下随机矩阵的谐波分析，揭示了经验谱分布与测度之间的马尔可夫-克雷因对应关系。

Construction of Liouville Brownian motion via Dirichlet form theory

The Liouville Brownian motion which was introduced in \cite{GRV} is a natural diffusion process associated with a random metric in two dimensional Liouville quantum gravity. In this paper we construct the Liouville Brownian motion via Dirichlet form theory. By showing that the Liouville measure is smooth in the strict sense, the positive continuous additive functional $(F_t)_{t \ge 0}$ of the Liouville measure in the strict sense w.r.t. the planar Brownian motion $(B_t)_{t \ge 0}$ is obtained. Then the Liouville Brownian motion can be defined as a time changed process of the planar Brownian mo…

正交对角映射上Dirichlet问题的多项式收敛速率

A Polynomial Rate of Convergence for the Dirichlet Problem on Orthodiagonal Maps

本文推广了先前工作，证明了正交对角映射上带Hölder边界数据的Dirichlet问题具有多项式收敛速率，关键方法是利用光滑磨光子使离散调和函数近似调和，并由此得到介观尺度上的Lipschitz性质。

Nodal Replication of Planar Random Waves

We study the almost periods of the eigenmodes of flat planar manifolds in the high energy limit. We prove in particular that the Gaussian Arithmetic Random Waves replicate almost identically at a scale at most ${\ell}$n := n -- 1 2 exp (Nn), where Nn is the number of ways n can be written as a sum of two squares. It provides a qualitative interpretation of the full correlation phenomenon of the nodal length, which is known to happen at scales larger than ${\ell}$ ' n := n --1/2 N A n. We provide also a heuristic with a toy model pleading that the minimal scale of replication should be closer t…

可整除性配置的衰减泊松狄利克雷逼近

Attenuated Poisson Dirichlet approximations for divisibility configurations

研究调和随机样本的不同素因子归一化对数点过程，证明其向Dickman泊松云原子序列（等价于均匀衰减泊松-狄利克雷律）的定量收敛。

Harmonic Functions And Linear Elliptic Dirichlet Problems With Random Boundary Values--Stochastic Extensions Of Some Classical Theorems And Estimates

Let $ψ:{\mathcal{D}}\rightarrow{\mathbf{R}}$ be a harmonic function such that $Δψ(x)=0$ for all $x\in\mathcal{D}\subset{\mathbf{R}}^{n}$. There are then many well-established classical results:the Dirichlet problem and Poisson formula, Harnack inequality, the Maximum Principle, the Mean Value Property etc. Here, a 'noisy' or random domain is one for which there also exists a classical scalar Gaussian random field (GRF) ${\mathscr{J}(x)}$ defined for all $x\in{\mathcal{D}}$ or $x\in\partial {\mathcal{D}}$ with respect to a probability space $[Ω,\mathcal{F},{\mathrm{I\!P}}]$. The GRF has vanishi…

The Dirichlet problem for orthodiagonal maps

We prove that the discrete harmonic function corresponding to smooth Dirichlet boundary conditions on orthodiagonal maps, that is, plane graphs having quadrilateral faces with orthogonal diagonals, converges to its continuous counterpart as the mesh size goes to 0. This provides a convergence statement for discrete holomorphic functions, similar to the one obtained by Chelkak and Smirnov for isoradial graphs. We observe that by the double circle packing theorem, any finite, simple, 3-connected planar map admits an orthodiagonal representation. Our result improves the work of Skopenkov and Wern…

The Laplacian on planar graphs and graphs on surfaces

These are lecture notes for the Current Developments in Mathematics conference at Harvard, November, 2011. We discuss topological, probabilistic and combinatorial aspects of the Laplacian on a graph embedded on a surface. The three main goals are to discuss: (1) for "circular" planar networks, the characterization due to Colin de Verdière of Dirichlet-to-Neumann operator; (2) The connections with the random spanning tree model; and (3) the characteristic polynomial of the Laplacian on an annulus and torus.