Large, Lengthy Graphs Look Locally Like Lines Itai Benjamini and Tom Hutchcroft May 9, 2019 Abstract We apply the theory of unimodular random rooted graphs to study the metric geometry of large, finite, bounded degree graphs whose diameter is proportional to their volume. We prove that for a positive proportion of the vertices of such a graph, there exists a mesoscopic scale on which the graph looks like R in the sense that the rescaled ball is close to a line segment in the Gromov-Hausdorff metric. 1 Introduction The aim of this modest note is to prove that large graphs with diameter proportional to their volume must `look like R' from the perspective of a positive proportion of their vertices, after some loc rescaling that may depend on the choice of vertex. We write dGH for the local Gromov-Hausdorff metric, a measure of similarity between locally compact pointed metric spaces that we define in detail in Section 1.1. Theorem 1.1. Let (Gn)n≥1 = ((Vn;En))n≥1 be a sequence of finite, connected graphs with jVnj ! 1, and suppose that there exists a constant C < 1 such that jVnj ≤ C diam(Gn) for every n ≥ 1. Suppose furthermore that the set of degree distributions of the graphs Gn are uniformly integrable. −1 Then there exists a sequence of subsets An ⊆ Vn with lim infn!1 jAnj=jVnj ≥ C such that loc lim sup inf dGH Vn; "dGn ; v ; ; d ; 0 = 0; n!1 ">0 R R v2An arXiv:1905.00316v3 [math.MG] 8 Nov 2020 where we write dGn for the graph metric on Gn and dR(x; y) = jx − yj for the usual metric on R. Here, the degree distributions of the graphs (Gn)n≥1 are said to be uniformly integrable if for every " > 0 there exists M such that P deg(v)1(deg(v) > M) ≤ "jV j for every n ≥ 1. v2Vn n This hypothesis holds in particular if there exists a constant M < 1 such that the graphs (Gn)n≥1 all have degrees bounded M. Our result should be compared to the (much more difficult) result of the first author, Finucane, and Tessera [2] that transitive graphs with volume proportional to their diameter converge to the 1 A scale of order r(n) A scale of order 1 s(n) r(n) Figure 1: A `comb' of length n with regularly spaced `teeth' of length r(n) with 1 r(n) n. On the scale r(n), the comb is metrically distinguishable from r(n) at every point. On scales s(n) satisfying r(n) s(n) n, the teeth of the comb are negligible and the comb looks like R from most of its points, with the exception of points near the left or right ends of the comb where the comb looks locally like R+ = [0; 1). On scales s(n) satisfying 1 s(n) r(n), the comb again looks like R from the perspective of most of its points, with the only exceptions being the points that are close to the top or bottom of a tooth (these bad points can be either on the teeth or the body of the comb). circle when rescaled by their diameter; here we have much weaker hypotheses but also a much weaker result. We remark that in [2] it is shown more generally that every sequence of transitive graphs with volume at most polynomial in their diameter has a subsequence converging to a torus (equipped with an invariant Finsler metric) when rescaled by their diameters. Various further results on the scaling limits of transitive graphs satisfying polynomial growth conditions have subsequently been obtained by Tessera and Tointon [8]. It seems unlikely that polynomial growth assumptions such C as jVnj = O diam(Gn) will imply anything of comparable strength to Theorem 1.1 about the metric geometry of graphs without the assumption of transitivity. Note that it is not possible to control the scale on which the rescaled graph looks like R. Indeed, if r : N ! N is any function with r(n)=n ! 0 and r(n) ! 1 as n ! 1, then the graph formed by attaching dn=r(n)e line segments of length br(n)c to a line segment of length n in a regularly spaced manner has diam(Gn) ∼ n and jVnj ∼ 2n but it metrically distinguishable from R on the scale r(n) from the perspective of every vertex in the sense that loc −1 lim inf dGH Vn; r(n) dGn ; v ; R; d ; 0 > 0: n!1 v2V R See Figure 1. Note also that the hypotheses do not allow us to take the sets An to have jAnj=jVnj ! 1. Indeed, consider taking an n × n square grid and attaching to this grid a path of length n2: The volume of this graph is about twice its diameter, the grid has about 1=2 of the total vertices of the 2 Figure 2: The graphs formed by attaching a line of length n2 to the corner of an n × n square grid, where n = 5; 20; 100. When n is large, the n × n grid is negligible from the perspective of the metric at the scales much larger than n but has a constant proportion of the total volume. If 2 we pick a point in the n × n grid uniformly at random, the graph will look locally like (R ; k · k1) 2 on scales 1 s(n) n and like R+ on scales n s(n) n with high probability. On scales of order n, the graph will look like a metric space formed by attaching a copy of R+ to the corner of ([0; 1]; λk · k1), where λ 2 (0; 1) depends on the precise choice of scale. From the perspective of a uniform random point in the line, which will not be near the ends of the line with high probability, 2 the graph will look locally like R on all scales 1 s(n) n . graph, and from a vertex of the grid the graph is metrically distinguishable from R at every scale. See Figure 2. (In this example, the graph instead looks locally like R+ from the perspective of the vertices of the grid. One can construct similar examples where the graph does not look locally like either R or R+ by attaching, say, three disjoint paths to the grid instead of one.) A similar example −1 shows that the dependence lim infn!1 jAnj=jVnj ≥ C is optimal. Unimodular random rooted graphs and Benjamini-Schramm limits. Although the state- ment is not probabilistic, the proof of Theorem 1.1 will rely on probabilistic arguments in an essential way. Roughly speaking, we will use the theory of Benjamini-Schramm limits to reduce Theorem 1.1 to a statement about certain probabilistic objects arising as subsequential limits in distribution of the sequence (Gn; ρn), where ρn is a uniform random vertex of Gn for each n ≥ 1. Let us now recall the relevant definitions, referring the reader to [5, 1] for more detailed treat- ments. A rooted graph is a connected, locally finite graph g together with a distinguished root vertex v. A graph isomorphism between rooted graphs is an isomorphism of rooted graphs if it preserves the root. The space of (isomorphism classes of) rooted graphs is denoted G•, and is equipped with the local topology, which is induced by the metric −R((g1;v1);(g2;v2)) dloc((g1; v1); (g2; v2)) = 2 where R((g1; v1); (g2; v2)) is the supremal value of r for which the balls of radius r around v1 and v2 are isomorphic as rooted graphs. The space G• is a Polish space (i.e., a topological space with a compatible complete metric) with respect to this topology [5, Theorem 2], and moreover is 3 separable since the (isomorphism classes of) finite rooted graphs form a countable dense subset. We will typically ignore the distinction between a rooted graph and its isomorphism class when this does not cause problems. Similarly, we define a rooted oriented-edge-labelled graph to be a rooted graph (g; v) to- gether with a function from the set of oriented edges of g to f0; 1g. (Although our graphs are f0;1g undirected, we can still think of each edge as a pair of oriented edges.) We write G• for the space of isomorphism classes of rooted oriented-edge-labelled graphs, which is equipped with a local topology that is defined similarly to the unlabelled case. Finally, we define the spaces of f0;1g doubly-rooted graphs G•• and doubly-rooted oriented-edge-labelled graphs G•• simi- larly to above except that we now have an ordered pair of distinguished root vertices. A probability measure µ on G• is said to be unimodular if it satisfies the mass-transport principle, which states that the identity1 2 3 2 3 X X µ 4 F (G; ρ, v)5 = µ 4 F (G; v; ρ)5 v2V v2V is satisfied for every measurable function F : G•• ! [0; 1], where we write (G; ρ) for a random f0;1g variable sampled from the measure µ. Unimodular probability measures on G• are defined similarly. We think of the function F as being an `automorphism equivariant' rule for sending non- negative amounts of mass between the different vertices of a graph g: The mass-transport principle says that for any such rule, the expected total mass the root sends out is equal to the expected total mass that the root receives.