An asymptotic expansion for the discrete harmonic potential

Random walks in the quarter plane, discrete harmonic functions and conformal mappings

We propose a new approach for finding discrete harmonic functions in the quarter plane with Dirichlet conditions. It is based on solving functional equations that are satisfied by the generating functions of the values taken by the harmonic functions. As a first application of our results, we obtain a simple expression for the harmonic function that governs the asymptotic tail distribution of the first exit time for random walks from the quarter plane. As another corollary, we prove, in the zero drift case, the uniqueness of the discrete harmonic function.

从超调和函数出发的离散高斯自由场Glauber动力学的混合剖面

Mixing profile for Glauber dynamics of the discrete Gaussian Free Field starting from super-harmonic functions

研究任意连通有限图上离散高斯自由场热浴Glauber动力学的混合时间，证明从超调和初始条件出发时具有强单调性，得到尖锐混合剖面，适用于无极端连通性的图，包括三维及以上网格有限盒子。

Constructing discrete harmonic functions in wedges

We propose a systematic construction of signed harmonic functions for discrete Laplacian operators with Dirichlet conditions in the quarter plane. In particular, we prove that the set of harmonic functions is an algebra generated by a single element, which conjecturally corresponds to the unique positive harmonic function.

On the sharpness of a three circles theorem for discrete harmonic functions

Any three circles theorem for discrete harmonic functions must contain an inherent error term. In this paper we find the sharp error term in an $L^2$-three circles theorem for harmonic functions defined in $\Zb^2$. The proof is highly indirect due to combinatorial obstacles and cancellations phenomena. We exploit Newton interpolation methods and recursive arguments.

Asymptotic Expansions for Moment Functionals of Perturbed Discrete Time Semi-Markov Processes

In this paper, we study mixed power-exponential moment functionals of nonlinearly perturbed semi-Markov processes in discrete time. Conditions under which the moment functionals of interest can be expanded in asymptotic power series with respect to the perturbation parameter are given. We show how the coefficients in these expansions can be computed from explicit recursive formulas. In particular, the results of the present paper have applications for studies of quasi-stationary distributions.

离散调和函数的阿尔姆格伦单调性公式

An Almgren monotonicity formula for discrete harmonic functions

研究无限组合图上调和函数的阿尔姆格伦单调性公式，通过证明其离散类似物，探讨了相关应用。

Discrete harmonic functions for non-symmetric Laplace operators in the quarter plane

We construct harmonic functions in the quarter plane for discrete Laplace operators. In particular, the functions are conditioned to vanish on the boundary and the Laplacians admit coefficients associated with transition probabilities of non-symmetric random walks. By solving a boundary value problem for generating functions of harmonic functions, we deduce explicit expressions for the generating functions in terms of conformal mappings. These mappings are yielded from a conformal welding problem with quasisymmetric shift and contain information about the growth of harmonic functions. Further,…

块计数约束调和和：谱展开与块定向欧拉-麦克劳林公式

Block-count constrained harmonic sums: spectral expansion and block-directed Euler-Maclaurin

本文精确刻画了固定数字块出现次数下调和和的性质，通过语言分解与随机基数展开建立谱理论，并给出介于欧拉-麦克劳林与泰勒展开之间的块定向插值公式。

球谐展开的稀疏恢复

Sparse recovery for spherical harmonic expansions

本文证明，通过球面上随机选取的少量样本，可以高效恢复稀疏球谐展开，主要方法是验证预处理随机测量矩阵的受限等距性质，并利用雅可比多项式一致增长的最新估计。

Discrete random walks on the group Sol

The harmonic measure $ν$ on the boundary of the group $Sol$ associated to a discrete random walk of law $μ$ was described by Kaimanovich. We investigate when it is absolutely continuous or singular with respect to Lebesgue measure. By ratio entropy over speed, we show that any countable non-abelian subgroup admits a finite first moment non-degenerate $μ$ with singular harmonic measure $ν$. On the other hand, we prove that some random walks with finitely supported step distribution admit a regular harmonic measure. Finally, we construct some exceptional random walks with arbitrarily small speed…

Quantitative estimates of discrete harmonic measures

A theorem of Bourgain states that the harmonic measure for a domain in $\R^d$ is supported on a set of Hausdorff dimension strictly less than $d$ \cite{Bourgain}. We apply Bourgain's method to the discrete case, i.e., to the distribution of the first entrance point of a random walk into a subset of $\Z ^d$, $d\geq 2$. By refining the argument, we prove that for all $\b>0$ there exists $ρ(d,\b)<d$ and $N(d,\b)$, such that for any $n>N(d,\b)$, any $x \in \Z^d$, and any $A\subset \{1,..., n\}^d$ $$ | \{y\in\Z^d\colon ν_{A,x}(y) \geq n^{-\b} \}| \leq n^{ρ(d,\b)}, $$ where $ν_{A,x} (y)$ denotes the…

Decay of harmonic functions for discrete time Feynman--Kac operators with confining potentials

We propose and study a certain discrete time counterpart of the classical Feynman--Kac semigroup with a confining potential in countable infinite spaces. For a class of long range Markov chains which satisfy the direct step property we prove sharp estimates for functions which are (sub-, super-)harmonic in infinite sets with respect to the discrete Feynman--Kac operators. These results are compared with respective estimates for the case of a nearest-neighbour random walk which evolves on a graph of finite geometry. We also discuss applications to the decay rates of solutions to equations invol…

Polynomials and harmonic functions on discrete groups

Alexopoulos proved that on a finitely generated virtually nilpotent group, the restriction of a harmonic function of polynomial growth to a torsion-free nilpotent subgroup of finite index is always a polynomial in the Mal'cev coordinates of that subgroup. For general groups, vanishing of higher-order discrete derivatives gives a natural notion of polynomial maps, which has been considered by Leibman and others. We provide a simple proof of Alexopoulos's result using this notion of polynomials, under the weaker hypothesis that the space of harmonic functions of polynomial growth of degree at mo…

A Karhunen-Loeve expansion for one-mode open quantum harmonic oscillators using the eigenbasis of the two-point commutator kernel

This paper considers one-mode open quantum harmonic oscillators with a pair of conjugate position and momentum variables driven by vacuum bosonic fields according to a linear quantum stochastic differential equation. Such systems model cavity resonators in quantum optical experiments. Assuming that the quadratic Hamiltonian of the oscillator is specified by a positive definite energy matrix, we consider a modified version of the quantum Karhunen-Loeve expansion of the system variables proposed recently. The expansion employs eigenvalues and eigenfunctions of the two-point commutator kernel for…

Harmonic pinnacles in the Discrete Gaussian model

The 2D Discrete Gaussian model gives each height function $η: \mathbb{Z}^2\to\mathbb{Z}$ a probability proportional to $\exp(-β\mathcal{H}(η))$, where $β$ is the inverse-temperature and $\mathcal{H}(η) = \sum_{x\sim y}(η_x-η_y)^2$ sums over nearest-neighbor bonds. We consider the model at large fixed $β$, where it is flat unlike its continuous analog (the Gaussian Free Field). We first establish that the maximum height in an $L\times L$ box with 0 boundary conditions concentrates on two integers $M,M+1$ with $M\sim \sqrt{(1/2πβ)\log L\log\log L}$. The key is a large deviation estimate for the …