An inequality for the heat kernel on an Abelian Cayley graph

We demonstrate a relationship between the heat kernel on a finite weighted Abelian Cayley graph and Gaussian functions on lattices. This can be used to prove a new inequality for the heat kernel on such a graph: when $t \leq t'$, $$\frac{H_t(u, v)}{H_t(u,u)} \leq \frac{H_{t'}(u, v)}{H_{t'}(u,u)}$$ This was an open problem posed by Regev and Shinkar.


Introduction
Let Γ = (V, E) be a finite undirected weighted graph. Let L be the graph laplacian of Γ. We then define the heat kernel H t as the matrix exponential e −tL , for any t ∈ R >0 . The value H t (u, v) can be thought of as the probability that a continuous-time random walk starting at u ends up at v after time t. See Section 1.1 of [RS16] for more detail on this interpretation. We will say that G has monotonic diffusion if the following holds whenever t ′ ≥ t, for all vertices v and u: We'll say that Γ is a weighted Abelian Cayley graph if there is an Abelian group structure on V under which Γ is translation-invariant. In this note we will prove the following theorem: Theorem 1.1. Any weighted Abelian Cayley graph has monotonic diffusion.
For arbitrary weighted undirected graphs, monotonic diffusion often fails to hold. See the appendix of [RS16] for a simple case. In 2013, Peres asked whether vertex-transitive graphs necessarily have monotonic diffusion. This was resolved in the negative by Regev and Shinkar in [RS16]. It also makes sense to ask whether a Riemannian manifold has monotonic diffusion, since there is a uniquely determined heat kernel in this setting, provided the manifold is complete and the Ricci curvature is bounded from below [C94]. We present an argument in Appendix A, due to Jeff Cheeger, that monotonicity does not always hold for Riemannian manifolds. In the other direction, it was shown in [RS15] that all flat tori have monotonic diffusion. Since weighted Abelian Cayley graphs are the closest thing we have to flat tori in the world of graph theory, it is natural to ask whether they have monotonic diffusion. This problem was posed in [RS16] and is resolved, in the affirmative, in this note.
We will use the following strategy: we show that, if υ is a nonnegative, even, real-valued function on a finite Abelian group, then the convolutional exponential exp * (υ) can be represented as a pushforward of a Gaussian function on a lattice. This will allow us to apply an inequality in [RS15] regarding Gaussian sums on lattice cosets.

Gaussian pushforwards
Throughout this paper, G will refer to an arbitrary finite Abelian group.
Definition 2.1. A lattice is a discrete subgroup of a finite-dimensional inner product space. If L is a lattice, we define the function ρ : L → R as ρ(x) = e −π x,x . When S ⊆ L, we'll use the notation ρ(S) to mean x∈S ρ(x).
Definition 2.2. We'll say that a function χ : G → R is a Gaussian pushforward if there exists a lattice L and a group homomorphism h : L → G such that χ(g) = ρ(h −1 (g)) for all g ∈ G. We'll use X to denote the set of all Gaussian pushforwards on G.
Proposition 2.3. X is closed under convolution.
Remark 2.4. Curiously, we also have that X is closed under multiplication, if we instead take L 3 to be the sublattice of Proposition 2.5. If χ ∈ X, then for all g 1 , g 2 ∈ G, we have Proof. This follows immediately from (4c) of corollary 2.2 of [RS15], if we take the lattice L to be the kernel of a homomorphism from a lattice to G whose corresponding Gaussian pushforward function is χ.
From continuity, we immediately have the following stronger statement: Corollary 2.6. If χ is in the topological closureX of X then for all g 1 , g 2 ∈ G, we have Proposition 2.7. If χ is inX, υ is a nonnegative, even, real-valued function on G, and ω = χ * υ, then for all g ∈ G, Proof. For all g ∈ G, we have Taking the mean of the first and third sums above, we have: Applying 2.6, Dividing by ω(0) yields the result.
3 The convolutional exponential of nonnegative even functions Definition 3.1. Suppose υ is a real-valued function on G. We define exp * (υ), the convolutional exponential of υ, by where υ * n is the nth convolutional power of υ. From the convolution theorem, we can equivalently define exp * (υ) by the equation where exp is the pointwise exponential operator. From this perspective, it's clear that exp In this section, we'll show that the convolutional exponential of any nonnegative even function on G is a Gaussian pushforward.
Definition 3.2. We'll use Φ to refer to the set of functions on G of the form φ(g) = δ(g − g 0 ) + δ(g + g 0 ) for some g 0 ∈ G. Here we have δ(0) = 1 and δ(x) = 0 for x = 0. We clearly have that Φ is a basis for the even functions on G.
Definition 3.3. We'll say that a sequence of real-valued functions υ n on G is O(f (n)) if, for any g ∈ G, we have that υ n (g) is O(f (n)). Since all norms on a finite-dimensional vector space are equivalent, this is the same as saying that υ n is O(f (n)), for any choice of norm on R G . We also have that υ n is O(f (n)) iffυ n is O(f (n)), by the Plancherel theorem. Proof. Without loss of generality, assume a ≥ b. We have: Since each term a i b n−1−i of the summation is at most C n−1 , we have: Since both sides are nonnegative, the above inequality proves the lemma.
Lemma 3.5. If a n and b n are both 1 + O(1/n), and a n − b n is O(f (n)), then a n n − b n n is O(nf (n)).
Proof. We have, for some C, that a n ≤ 1 + C/n and b n ≤ 1 + C/n. By applying 3.4, we have Lemma 3.6. Suppose φ ∈ Φ and α ∈ R. Then we can find a sequence of Gaussian pushforwards (χ n ) such that δ + αφ/n − χ n is O(1/n 4 ).
Proof. We'll choose χ n arbitrarily when n ≤ α, and the rest of this proof will be concerned with the tail (χ n ) n>α . Choose a g 0 such that φ(g) = δ(g + g 0 ) + δ(g − g 0 ) for all g ∈ G. Let r n = ln(n/α)/π, so that α/n = ρ(r n ). Let L n be the lattice in R of reals of the form kr n , with k ∈ Z. We then have a unique group homomorphism h n from L n to G that sends r n to g 0 . We define χ n by χ n (g) = ρ(h −1 n (g)). Let A n = {r n , 0, −r n } and B n = L n \ A n . We can break χ n into two smaller sums: The left term is equal to δ + αφ/n. The right term is O(1/n 4 ). This yields the result.
Proof. Let ψ = exp * (αφ). From the convolution theorem, we have thatψ = exp(αφ). Therefore, we haveψ From 3.6, we have a sequence of Gaussian pushforwards (χ n ) such that δ + αφ/n − χ n is O(1/n 4 ). Taking the Fourier transform, we have that 1 + αφ/n −χ is O(1/n 4 ). We then have from 3.5 that (1 + αφ/n) n −χ n n is O(1/n 3 ). Combining this with the formula forψ above, we have: Applying the convolution theorem then yields: From 2.3, we know that χ * n n is a Gaussian pushforward, so this last limit proves the result.
Theorem 3.8. Suppose υ is a nonnegative even function on G. Then exp * υ is inX.
Proof. We can clearly represent υ as a sum α 1 φ 1 + · · · + α n φ n , with α i being positive scalars and φ i ∈ Φ. Then, applying the convolution theorem, we can represent exp * (υ) as the convolution exp * (α 1 φ 1 ) * · · · * exp * (α n φ n ). From 3.7, we know that each individual term in this convolution is in the closure of X, so the result follows from 2.3 and the continuity of the convolution operation.

The heat kernel on weighted Abelian Cayley graphs
In this section, we prove 1.1. The hard work has already been done in [RS15] and in the previous section, now we just need to translate these results into the language of graph theory.
Proof of Theorem 1.1. We define τ to be the element of R V obtained by applying the graph laplacian to −δ. In other words, we have τ : V → R and τ (v) = −L(0, v), with L the graph laplacian. Then τ is an even function and is nonnegative everywhere except at 0. We also have that the linear operator determined by the graph laplacian is just convolution with −τ ; it follows that the heat kernel matrix H t corresponds to convolution with exp * (tτ ), and therefore H t (u, v) = exp * (tτ )(u − v). So, to prove 1.1, it suffices to show that, for t ≤ t ′ , and for any v ∈ V , Let τ ′ : V → R be given by τ ′ (0) = 0 and τ ′ (v) = τ (v) for v = 0. Then τ ′ is a nonnegative even function. We also have that exp * (tτ ′ ) = e −τ (0) exp * (tτ ). It therefore suffices to show that, for t ≤ t ′ , and for any v ∈ V , , the result follows immediately from 3.8 and 2.7.

Directions for further work
In order to prove 1.1, we needed the inequality 2.6 for χ a Gaussian pushforward: which itself follows from the stronger inequality, proven in [RS15]: The proof of monotonic diffusion on flat tori also uses (1), except χ(g) is replaced by H t (0, g), with H the heat kernel on the torus in question.
So, if we want to see where else we have monotonic diffusion, it is natural to ask: where else does (1) hold? On the surface, it looks we need group structure to even state (1). However, (1) can be reformulated in the language of Riemannian symmetric spaces. Indeed, suppose M is a Riemannian symmetric space, that is, M is a connected Riemannian manifold equipped with an isometric automorphism s p for each point p, such that s p (p) = p and the derivative of s p at p is the negation map on T p . For flat tori then, we have a symmetric space structure with s x (y) = 2x − y, and so the following is equivalent to (1): To see this, substitute 0 for a, x for b, and x + y for c. Then s b (c) is x − y, and we recover the original inequality.
It is shown Appendix B that any symmetric space which satisfies (2) has monotonic diffusion. This inequality doesn't hold for arbitrary symmetric spaces; it is shown in Appendix C that it fails on H 3 . However, we already know from [RS15] that (2) holds for flat tori, and the author has numerically tested (2) for the 2-sphere and real projective 2-space, using the spherical harmonic functions of SciPy to approximate the heat kernel. The inequality appears to hold for these spaces. It seems natural then to ask: for which Riemannian symmetric spaces does (2) hold? Perhaps it holds on all Riemannian symmetric spaces of compact type.
It can also be readily seen from equation (1.6) of [GN98] that the heat kernel has monotonic diffusion on H 3 , despite the fact that (2) fails on this space. Perhaps monotonic diffusion holds on other hyperbolic spaces as well.

Ackowledgements
The train of thought that lead to the main idea of this paper began with a discussion with Alexander Borisov. He had noticed that strongly positive-definite functions obey the main inequality of [RS15]. It was through trying to relate this to the fact that Gaussian pushforwards obey the same inequality that I came across the construction used in section 3. So I thank him for the stimulating discussion, as well as for helpful comments on a draft. I thank Oded Regev as well for helpful comments and discussion. This paper was written while I was visiting the University of Münster. I thank Christopher Deninger for making this possible.