Unique continuation on planar graphs

Unique continuation on planar graphs, Discrete Analysis 2025:16, 12 pp.

A fundamental result in harmonic analysis is Liouville’s theorem for harmonic functions: that a bounded harmonic function on $\mathbb R^n$ is constant. It is of course closely related to Liouville’s theorem from complex analysis, that a bounded holomorphic function on $\mathbb C$ is constant.

An active area of more modern research, to which this paper contributes, is the study of discrete analogues of such theorems. For example, one can consider the graph $\mathbb Z^d$, with edges from $x$ to $x\pm e_i$ for each $i$, and define a function $f:\mathbb Z^d\to\mathbb R$ to be harmonic if its discrete Laplacian is zero – or equivalently, if for every vertex $x$, $f(x)$ is equal to the average of $f$ over all neighbours of $x$. Blackwell proved in 1955 that $\mathbb Z^d$ has the Liouville property: that is, bounded harmonic functions on $\mathbb Z^d$ are constant.

In 2017, Lev Buhovsky, Alexander Logunov, Eugenia Malinnikova and Mikhail Sodin proved a rather remarkable strengthening of this theorem when $d=2$. It turns out that one does not need to assume that $f$ is bounded everywhere, but only that it is bounded on a sufficiently large part of $\mathbb Z^2$. More precisely, they showed that there is a constant $\epsilon>0$ such that every harmonic function that is bounded on a set $X$ that intersects all sufficiently large sets $[-N,N]^2$ in a subset of density at least $1-\epsilon$ must be constant. (Examples show that $\epsilon$ must be less than 1/2, but determining the best $\epsilon$ is still an open problem. There are also examples that show that the corresponding strengthening when $d>2$ is false.)

To prove their result, they first obtained an exponential lower bound for the growth of a non-constant harmonic function that is bounded on most of $\mathbb Z^2$. They then obtained an exponential upper bound, which for sufficiently small $\epsilon$ was smaller than the lower bound, which gave a contradiction unless for that $\epsilon$ every function bounded on a $1-\epsilon$ proportion of $\mathbb Z^2$ is constant.

The purpose of this paper is to extend the results of Buhovsky, Logunov, Malinnikova and Sodin to more general periodic planar graphs and more general Laplacians. A planar graph is said to be periodic if it can be embedded into $\mathbb R^2$ in such a way that it is invariant under a set of translations from some 2-dimensional lattice. The paper also allows replacing the average over neighbours by a weighted average, as long as the weights used are also invariant under the lattice translations. Examples show that the planarity condition is needed here.

It turns out that the proof of the exponential lower bound from the $\mathbb Z^2$ case (with the standard Laplacian) carries over fairly straightforwardly to this more general context, though the proof given here is somewhat simpler. However, the proof of the upper bound makes essential use of features, in particular a polynomial structure for functions that are constant along two parallel diagonals, that are delicate to prove and particular to $\mathbb Z^2$, so the authors of this paper needed to come up with a completely different proof. The proof they have found is a considerably simpler argument that is topological in nature and therefore much more robust to changes in the set-up of the problem.

The rough idea is as follows. They first consider functions that are not just small on most of the graph but are actually zero on most of it. By averaging, such a function must be zero on most of the boundary vertices of arbitrarily large balls. From that the authors conclude that the function must be identically zero, since otherwise one could find a non-zero point deep inside a large ball. If the value is positive, say, then by the maximum principle, the positive connected region containing that point must stretch all the way to the boundary, forming a long thread. But since the graph is planar, and since any zero that is adjacent to a positive value must also be adjacent to a negative value, there must be many disjoint long threads of this kind with alternating signs. This forces the function to be mostly non-zero on the boundary, a contradiction. The argument can then be modified to obtain a Liouville theorem under the weaker assumption that a function is small on almost all the boundary.