[Paper Review] On-diagonal lower estimate of heat kernel on graphs

Yong Lin, Yiting Wu|arXiv (Cornell University)|Dec 27, 2016

Geometric Analysis and Curvature Flows14 references4 citations

TL;DR

This paper establishes a new on-diagonal lower estimate for the heat kernel on graphs for large times using only a volume growth condition, without requiring curvature-like assumptions. By first deriving an upper bound in the natural graph metric and then applying volume growth, it proves a lower bound of the form $ p(t,x,x) \geq \frac{c}{V(x,\sqrt{t})} $, extending previous results under weaker conditions than curvature-based methods.

ABSTRACT

The purpose of this paper is to establish a new continuous-time on-diagonal lower estimate of heat kernel for large time on graphs. To achieve the goal, we first give an upper bound of heat kernel in natural graph metric, and then use this bound and the volume growth condition to show the validity of the on-diagonal lower bound.

Motivation & Objective

• To derive a lower bound for the on-diagonal heat kernel on graphs under minimal geometric assumptions.
• To avoid reliance on curvature conditions like $ CDE(n,0) $ or $ CDE'(n,0) $, which constrain the graph structure.
• To establish the lower bound using only the volume growth condition $ V(x,r) \leq c r^m $, a more general and accessible assumption.
• To extend the applicability of heat kernel estimates to graphs with polynomial volume growth, even when curvature conditions do not hold.
• To provide a continuous-time heat kernel estimate that is valid for large time, using a natural graph metric.

Proposed method

• Derive an upper bound for the heat kernel $ p(t,x,y) $ in the natural graph metric $ d(x,y) $, using techniques inspired by Davies and Folz.
• Use the upper bound to control the tail of the heat kernel distribution outside a ball $ B(x,r) $, estimating the mass outside $ B(x,r) $.
• Split the complement of $ B(x,r) $ into dyadic annuli $ B(x,2^{k+1}r) \setminus B(x,2^k r) $ to bound the sum over distant vertices.
• Apply the volume growth condition $ V(x,r) \leq c r^m $ to bound the measure of annuli and control the exponential decay in the upper bound.
• Choose $ r(t) = Ct \log t $ with $ C > 2D_{\mu}e $ to ensure the tail sum decays to zero as $ t \to \infty $, proving the on-diagonal lower bound.
• Use the fact that the total mass of the heat kernel is 1 and the tail mass is less than $ 1/2 $ for large $ t $, so the on-diagonal mass must be bounded below.

Experimental results

Research questions

• RQ1 Can a non-Gaussian on-diagonal lower bound for the heat kernel on graphs be established under only a volume growth condition, without curvature assumptions?
• RQ2 What is the sharpness of the lower bound $ p(t,x,x) \geq c / V(x,\sqrt{t}) $ in the context of large-time heat kernel behavior on graphs?
• RQ3 How does the natural graph metric influence the derivation of heat kernel estimates compared to transformed metrics like $ d_\theta $?
• RQ4 To what extent can the DGG Lemma and related techniques be adapted to derive lower bounds via upper bound control?
• RQ5 Can the tail mass of the heat kernel be controlled sufficiently to infer a lower bound on the diagonal value?

Key findings

• The paper establishes a lower bound of the form $ p(t,x,x) \geq \frac{c}{V(x,\sqrt{t})} $ for large $ t $, valid under the polynomial volume growth condition $ V(x,r) \leq c r^m $.
• The upper bound for the heat kernel is derived in the natural graph metric $ d(x,y) $, avoiding the need for metric transformations used in prior work.
• The tail sum $ \sum_{z \in B(x,r)^c} \mu(z) p(t,x,z) $ is shown to decay to zero as $ t \to \infty $ when $ r(t) = Ct \log t $ with $ C > 2D_{\mu}e $, ensuring the on-diagonal mass is bounded below.
• The decay rate of the tail is controlled via dyadic annuli and exponential decay terms involving $ \log $, leading to a sum that vanishes in the limit.
• The result holds for all $ t \geq T $, where $ T $ is sufficiently large, and the lower bound is uniform across all vertices $ x \in V $.
• The method avoids curvature-based assumptions like $ CDE'(n,0) $, making the result applicable to a broader class of graphs with only volume growth.