Dmitry Chelkak

Robust discrete complex analysis: A toolbox

We prove a number of double-sided estimates relating discrete counterparts of several classical conformal invariants of a quadrilateral: cross-ratios, extremal lengths and random walk partition functions. The results hold true for any simply connected discrete domain Ω with four marked boundary vertices and are uniform with respect to Ω’s which can be very rough, having many fiords and bottlenecks of various widths. Moreover, due to results from [Boundaries of planar graphs, via circle packings (2013) Preprint], those estimates are fulfilled for domains drawn on any infinite “properly embedded” planar graph Γ⊂C (e.g., any parabolic circle packing) whose vertices have bounded degrees. This allows one to use classical methods of geometric complex analysis for discrete domains “staying on the microscopic level.” Applications include a discrete version of the classical Ahlfors–Beurling–Carleman estimate and some “surgery technique” developed for discrete quadrilaterals.

The inverse problem for perturbed harmonic oscillator on the half-line with a dirichlet boundary condition

Abstract.We consider the perturbed harmonic oscillator

Discrete complex analysis on isoradial graphs

T_{D}\psi=-\psi^{\prime\prime}+x^{2}\psi+q(x)\psi, \psi(0)=0,

Crossing Probabilities in Topological Rectangles for the Critical Planar FK-Ising model

in

Weyl–Titchmarsh functions of vector-valued Sturm–Liouville operators on the unit interval

L^{2}(\mathbb{R}_{+})

Discrete stress-energy tensor in the loop O(n) model

, where