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Heat kernel asymptotics for scaling limits of isoradial graphs

Simon Schwarz, Anja Sturm, Max Wardetzky

Abstract

We consider the asymptotics of the discrete heat kernel on isoradial graphs for the case where the time and the edge lengths tend to zero simultaneously. Depending on the asymptotic ratio between time and edge lengths, we show that two different regimes arise: (i) a Gaussian regime and (ii) a Poissonian regime, which resemble the short-time asymptotics of the heat kernel on (i) Euclidean spaces and (ii) graphs, respectively.

Heat kernel asymptotics for scaling limits of isoradial graphs

Abstract

We consider the asymptotics of the discrete heat kernel on isoradial graphs for the case where the time and the edge lengths tend to zero simultaneously. Depending on the asymptotic ratio between time and edge lengths, we show that two different regimes arise: (i) a Gaussian regime and (ii) a Poissonian regime, which resemble the short-time asymptotics of the heat kernel on (i) Euclidean spaces and (ii) graphs, respectively.
Paper Structure (9 sections, 13 theorems, 108 equations, 1 figure)

This paper contains 9 sections, 13 theorems, 108 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Isoradial graphs
  4. Geometric Graph Laplacian
  5. Discrete Heat Kernels
  6. Euclidean Limit
  7. Large deviation principle
  8. Gaussian lower bound
  9. Graph Regime

Key Result

Lemma 2.1

There is a constant $\kappa \leq 1.998$ such that for all $h>0$ and $u,v\in V_h$ If additionally Assumption ass:bap-primal holds, then for $h>0$ and $u,v\in V_h$, where $c_p$ is the constant in Assumption ass:bap-primal.

Figures (1)

  • Figure 1: Vertex fan in an isoradial graph. Left: All faces are inscribed into circles of equal diameter. Middle: Dual edge $e^*$ defined as the segment between midpoints of circumcircles of two adjacent polygons. The resulting polygon enclosed by dual edges is the dual face of $u$ with area $A_u$. Right: Resulting rhombic tiling of the plane.

Theorems & Definitions (29)

  • Lemma 2.1
  • proof
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • ...and 19 more