Edge length dynamics on graphs with applications to $p$-adic AdS/CFT

We formulate a Euclidean theory of edge length dynamics based on a notion of Ricci curvature on graphs with variable edge lengths. In order to write an explicit form for the discrete analog of the Einstein-Hilbert action, we require that the graph should either be a tree or that all its cycles should be sufficiently long. The infinite regular tree with all edge lengths equal is an example of a graph with constant negative curvature, providing a connection with $p$-adic AdS/CFT, where such a tree takes the place of anti-de Sitter space. We compute simple correlators of the operator holographically dual to edge length fluctuations. This operator has dimension equal to the dimension of the boundary, and it has some features in common with the stress tensor.


Introduction
Dynamical geometry in the bulk of anti-de Sitter space is a cornerstone of the study of the anti-de Sitter / conformal field theory correspondence (AdS/CFT). At the linearized level, propagation of gravitons in AdS can be translated into the two-point function of the stress-energy tensor in the CFT. At the non-linear level, dynamical geometry is involved in everything from anomalies to holographic renormalization group flows to the formation of black holes.
Recent developments [1,2] in the study of holographic relations between field theories defined on the p-adic numbers and bulk dynamics defined on a regular tree graph have omitted the study of dynamical geometry in the bulk. Different bulk topologies were considered in [2] in connection with non-archimedean generalizations of BTZ black holes, following earlier work [3]; but it has generally been assumed that all edges and all vertices on the tree are locally indistinguishable. In this paper, we want to lift this restriction by considering variable edge lengths. More specifically, we start with an action on the tree of the form Here xy indicates a sum over edges (i.e. without counting xy and yx separately), and a xy is the length of the edge xy, while V is a potential for the bulk scalar field φ x .
Calculations of correlators of the operator dual to φ x were carried out in [1,2] with all a xy set equal to 1, and these calculations have notable precursors in the literature on p-adic strings, for example [4]. 1 Now we would like to ask what interesting dynamics for the edge lengths a xy could be added. 2 To get started, let's set where e = xy is an edge. Then J e is a "bond strength" or "exchange energy" for the edge e.
All our discussion focuses on Euclidean signature, in which all the bond strengths are positive.
One obvious way to make the bond strengths dynamical is to include some Gaussian white noise in the J e : that is, we could draw each J e independently from a Gaussian distribution.
White noise for the J e seems quite unlike gravitational dynamics, because nearby J e don't pull on one another. Better would be to introduce some interactions among the J e on neighboring edges by adding to the (1) an action where ef means a sum over neighboring edges-that is, edges which share one vertex. If we omitted the first term in (3), and made the potential U quadratic, then the J e would be independent from one another, and we would be back to the case of Gaussian white 1 Meanwhile, an apparently different approach to dynamics on the tree was advanced in [5], in which a directed structure on the graph is assumed, such that each vertex has a single parent and p offspring. Then one defines a process that probabilistically assigns the state of each vertex based only on the state of its parent. Holographic correlators can be constructed in this approach in terms of the limits of combinations of the probabilities of vertices which are many steps down along the tree. 2 Of course, one could imagine also introducing some dynamics for parameters in the potentials in V that vary from vertex to vertex, but since this could be done simply by adding another field θ x on vertices and introducing θ-φ interactions, we don't think of it as such an interesting avenue. noise (but unquenched assuming we form a partition function Z = DJDφ e −S φ −S J ). In particular, we see that a quadratic term in the U corresponds to a mass term for the edge variables J e . Probably for something resembling gravity, we should avoid having a quadratic term in the U .
While (3) is a sensible starting point, it seems ad hoc. A key idea that will lead us to a more interesting class of edge length actions is a notion of Ricci curvature on graphs with variable edge lengths. Closely related ideas have been developed in the mathematical literature for some time: see for example [6,7,8]. Our main point of departure is the definition of Ricci curvature in [7,8] as a function of pairs of vertices (not necessarily neighboring vertices), based on a comparison of distance between the two chosen vertices and a weighted distance between two probability distributions, each one localized near one of the chosen vertices. Our extension of this notion of Ricci curvature to the case of variable edge lengths has some arbitrariness, so we cannot claim to have a uniquely privileged definition of the graph-theoretic Ricci curvature. However, we do have a well motivated class of constructions with good properties, including the finding that the regular tree graph with all edge lengths equal has constant negative curvature.
The plan of the rest of this paper is as follows. In section 2.1 we briefly review the connection between the p-adic numbers and the regular tree graph with coordination number p + 1. Then in section 2.2 we explain how the action (3) leads to a notion of edge Laplacian which is different from the usual one, but natural from the point of view of the so-called line graph. Next, in section 3, we give the definition of Ricci curvature which we will use. While our motivation is p-adic AdS/CFT, edge length fluctuations can be studied on more general graphs. The particular Ricci curvature construction that we introduce depends on the graph being "almost a tree," in a sense that we will make precise in section 3. (Intuitively, what "almost a tree" means is that all cycles in the graph should be sufficiently long.) We explain in section 3.1 how a linearized analysis around the regular tree reduces the Ricci curvature to the edge Laplacian of the bond strengths J xy . We exhibit in section 3.2 an analog of the Einstein-Hilbert action, with a boundary term similar to the Gibbons-Hawking action.
This action leads to equations of motion which are satisfied by the regular tree with equal edge lengths, and the linearized fluctuations are controlled as expected by the edge length Laplacian. We compute in section 4 the simplest holographic correlators involving edge length fluctuations. In section 5 we describe an exact solution to the equations of motion on a regular tree which deviates strongly from constant edge length. We conclude in section 6 by reviewing our main results and indicating some direction for future work. Appendix A reviews aspects of the action of the p-adic conformal group on the graph whose boundary is the p-adic numbers. Appendix B explains the Vladimirov derivative, which is a crucial construction in p-adic field theory and was understood in the context of bulk reconstruction [2] to be effectively a normal derivative at the boundary of the tree.

Mathematical background
In this section we briefly review two well-known mathematical concepts. In subsection 2.1 we explain the Bruhat-Tits tree, a regular tree whose boundary is the p-adic numbers. In subsection 2.2 we summarize the line graph construction, which renders natural the edge Laplacian that we encounter when linearizing the graph theoretic Ricci curvature to be introduced in section 3.

p-adic numbers and the Bruhat-Tits tree
Introductions to p-adic numbers requiring a minimum of technical background can be found in the recent works [1,2] and in the earlier literature on p-adic string theory, notably [9].
Here we sketch only a few of the most relevant points.
For any chosen prime integer p, the p-adic numbers Q p are the completion of the rationals Q with respect to the p-adic norm, defined on Q so that if a and b are non-zero integers, neither of which is divisible by p, then By definition, |0| p = 0. We will usually drop the subscript p and write |x| instead of |x| p when it is obvious from context that we mean the p-adic norm. The p-adic norm is ultrametric, meaning that |x+y| ≤ max{|x|, |y|}. Q p is a field, with multiplication, addition, and inverses defined by continuity from their usual definitions on Q.
Any non-zero p-adic number can be expressed uniquely as a series: where v ∈ Z, c 0 ∈ F × p , and c i ∈ F p . Here F × p denotes the non-zero elements in F p . 3 The infinite series in (5) appears to be highly divergent, but in fact it converges because the c i are bounded in p-adic norm, while |p n | = p −n . The expansion (5) is reminiscent of the base p representation of a real number, but it is different because it terminates to the right and may continue indefinitely to the left.
The Bruhat-Tits tree, which we denote T p , can be understood informally as a graphical representation of the expansion (5). We picture an infinite regular tree with coordination number p + 1, with a privileged path leading through it (with no back-tracking) from a boundary point that we label ∞ to another boundary point that we label 0. We describe this privileged path as the "trunk" of the tree. We now consider another path (also with no back-tracking) starting from the point ∞ and leading to some other boundary point that we are going to associate with the p-adic number x. This new path must run along the trunk for a while, and the location where it diverges from the trunk can be labeled by the valuation v of x (as it appears in (5)). When we branch off the main trunk, the first step we take requires a choice out of p − 1 possible directions, so we can label this choice by an element In each subsequent step, we have to choose among p possible directions, and each such choice can be labeled by an element c i ∈ F p . In short, we see that the data required to select the new path is in precise correspondence with the information required to specify a non-zero p-adic number. Since infinite non-back-tracking paths from ∞ through the tree are in precise correspondence with the boundary points other than ∞, we can say that the boundary of the tree is Q p ∪ {∞}, which is P 1 (Q p ). 4 It can be shown that the Bruhat-Tits tree is a quotient space: where Z p denotes the p-adic integers (the completion of Z with respect to |·| p , or equivalently the set of all x ∈ Q p with |x| p ≤ 1). The quotient (6) is similar to the realization of the Poincaré disk as SL(2, R)/U(1). A similar construction can be given for field extensions of the p-adic numbers: for example, the unramified extension of degree n, which we denote Q q with q = p n , is associated with a tree T q = PGL(2, Q q )/PGL(2, Z q ) with coordination number p n + 1. Non-zero elements x ∈ Q q admit an expansion of the form (5), except that the finite field F p is replaced by the larger finite field F q . Having made such an expansion, the norm of x can be defined by |x| = p −v .
The action of PGL(2, Q q ) on a number x ∈ Q q is realized through linear fractional transformations, and in particular it includes scaling x by any integer power of p. Consider scaling by p m for some m > 1. This corresponds to an isometry of T q based on a translation along the main trunk of the tree by m steps. The group Γ generated by this translation and its inverse is an image of Z inside PGL(2, Q q ), and the quotient space T q /Γ is analogous to the construction of the BTZ black hole as a quotient by some subgroup Γ ⊂ SO(3, 1) of the three-dimensional hyperbolic plane H 3 = SO(3, 1)/SO(3). By construction, T q /Γ has a single cycle with m links, and otherwise its structure is that of a regular tree. It is possible to consider more complicated groups Γ, and this is precisely the direction explored in [3,2].
It is also possible to consider more general extensions of Q p than the unramified extension Q q , but we leave an explicit account along such lines for future work.

An edge Laplacian
Consider the action (3) on a graph G. For applications to p-adic AdS/CFT, G should be the Bruhat-Tits tree T q or something close to it, but all of what we will say in this section applies to a general, connected, undirected graph G, provided no edge of G can have both its ends on the same vertex, and between any two vertices of G there is at most one edge.
It is easy to check that the equation of motion for J following from the action (3) is where we define an edge Laplacian as Here

Ricci curvature on graphs
While the action (3) seems natural enough from the point of view of dynamical models on graphs, we would prefer to have some geometrical starting point that would allow us to identify a graph-theoretic analog of the Einstein-Hilbert action. At first it seems like a hopeless task to construct such an action on a tree graph, because the Einstein-Hilbert action involves the Ricci scalar R, which is usually constructed as a contraction of the Riemann tensor R µν α β . But R µν α β is generally thought of as the field strength of the Christoffel connection; in other words, it describes holonomies around small loops. With no loops, it's hard to see how to define non-trivial field strengths. To avoid this, we want to take advantage of constructions of analogs of the Ricci tensor R µν that do not depend on connections at all, but instead on some notion of transport distance.
To build intuition, let's recount a standard result (see for example [10]) that goes in the direction we want, but which is framed in the context of a smooth D-dimensional manifold with a Euclidean metric which induces a distance function d(x, y) between any two points on the manifold. Given two points x 0 and y 0 , separated by a small distance r, choose some much smaller distance a r and consider balls B x 0 and B y 0 , comprising all points x with d(x, x 0 ) < a and all points y with d(y, y 0 ) < a, respectively. Let n µ be the unit vector in the direction from x 0 to y 0 ; we are not concerned with exactly which tangent space n µ lies in because we wish to use it in an asymptotic formula which can absorb O(r) uncertainties in n µ . Likewise we consider the Ricci curvature R µν at x 0 or y 0 , or anywhere within a radius r of either of these points. There is a natural way to define a transport distance W (B x 0 , B y 0 ) between the two balls; essentially it is a weighted distance of separations of points in B x 0 and B y 0 , but we postpone its precise definition. Then we can form a bilocal quantity The second equality in (9) is the result we are interested in. It tells us that the leading behavior of κ(x 0 , y 0 ) for small a and r contains all the information in R µν -provided we are allowed to know κ(x 0 , y 0 ) for all possible directions of separation n µ . See figure 2.
x 0 y 0 r a a n μ x 0 y 0 ψ x 0 (t) Figure 2: Left: Small spherical neighborhoods of nearby points in a smooth manifold provide a starting point for defining Ricci curvature without first defining the Riemann tensor. Right: A similar construction on graphs hinges on replacing the small spherical neighborhood around a point x 0 with a probability distribution ψ x 0 (t) which for small t is concentrated at x 0 with a little bit of weight on neighboring vertices. Now let's return to the definition of the transport distance W appearing in (9). Consider the so-called Wasserstein distance W (p 1 , p 2 ) between two probability measures on our smooth manifold. We introduce the set L 1 of 1-Lipschitz functions, which are real-valued function on our smooth manifold satisfying for all x and y.
Then the Wasserstein distance is Having defined W on probability measures, we define it on unit balls B x 0 and B y 0 by replacing each ball by the uniform probability distribution supported on the ball. To , which to a first approximation takes the form where n µ = g µν n ν and g µν is the Euclidean metric tensor.
When it comes to graphs, our first impulse might be to require two points x 0 and y 0 to be separated by r 1 steps and then consider something similar to the definition (9) with the balls replaced by the nearest neighbors of x 0 and y 0 . This is unattractive because our eventual aim is for κ(x 0 , y 0 ) to be defined for neighboring x 0 and y 0 , so that κ(x 0 , y 0 ) can be thought of as defined on each edge; and then we hope to find in some sort of linearized analysis that κ on edges is closely related to the edge Laplacian of fluctuations j xy in the bond strengths, similar to the way the Ricci tensor on a nearly flat manifold is related to the Laplacian of the metric. So, how do we find some construction on a graph resembling a ball whose radius is much smaller than the length of a single edge?
The answer of [7,8] (with closely related ideas appearing in [7]) is to consider for a fixed vertex x 0 a probability distribution ψ x 0 (x, t) with most of its weight at x = x 0 and a small amount of weight at neighboring vertices, so that the average distance from x 0 of a vertex chosen from this distribution is much less than an edge length. More precisely, for sufficiently small positive real t, we set We have defined and, as always, we require J xy = 1/a 2 xy for all edges. The factor of d J (x 0 ) in (12) ensures that ψ x 0 (x, t) is a probability distribution. As is evident from the definition, D x 0 is a sort of lapse function which tells us how fast the "time" t runs at different locations on the graph.
Clearly, the definition (12) is closely connected to a diffusive process. To make this connection more precise, consider the vertex Laplacian If we define a diagonal matrix on edges, Λ ee = J e δ ee , then it is easy to show that = d † Λd, and by inspection If we want our constructions to reduce to those of [8] in the case when all the edge lengths a xy = 1/ J xy are equal to 1, then we should set D x 0 to be equal to the degree of the vertex With the probability distributions ψ x 0 (x, t) in place, we can follow the spirit of (9) precisely. First we define a distance function on the graph d(x, y) as the minimum possible sum of edge lengths a e along a path connecting x and y. Then 1-Lipschitz functions f (x) defined on vertices are precisely the functions satisfying the inequality (10), and (11) is trivially Following [7,8] (with variable edge lengths), we define What we mean by ψ x (t) is the probability distribution ψ x (t,x) for all verticesx on the graph.
It is illuminating now to compute κ(x, y) for x and y on opposite ends of an edge in a 5 Recent related work [11,12] on Ricci curvature of weighted graphs starts with a Laplacian = − 1 , which is suggestive of the choice D x = d J (x). But it is hard to make a precise comparison with our work since much of the development in [11,12] follows [6] rather than [7,8]; also, the focus in [11,12] is on estimation of eigenvalues of , and the graphs of interest are usually those with non-negative Ricci curvature, whereas we are mostly interested in negative curvature. tree graph. As we go through the calculation, we will see that it can be extended to graphs whose cycles are sufficiently long, in a sense that we will make precise. We do not require for the following computation that the graph should be the Bruhat-Tits tree, but this is of course what we have in mind eventually in order to connect to p-adic AdS/CFT. What makes the tree graph computation straightforward is that we can easily see what the supremizing 1-Lipschitz function f should be. Let x i be the vertices adjacent to x other than y, and let y i be the vertices adjacent to y other than x. Then we can set An additive constant in f doesn't affect the Wasserstein distance, so setting f (x) = 0 is just a convention. The other choices are designed to make f as positive as possible in the region where ψ x (t) has most of its weight, and as negative as possible in the region where ψ y (t) has most of its weight. We cannot do better than (18) because f already saturates the inequality (10) for pairs of points which are ordered in the sense of the partial ordering x i x y y i .
If our graph is not a tree, then there is the possibility that some x i might be connected to some y i by a path which is shorter (in the sense of sums of edge lengths) than the path that leads through the edge xy-and if that were so, then no 1-Lipschitz function could have the values indicated in (18). In order to prevent such a situation, it is sufficient to require that the graph should have no cycle with fewer than seven edges, and that the variation in edge lengths within a given cycle is by no more than a factor of 4/3. 6 Then it is guaranteed that no path between an x i vertex and a y i vertex can be shorter than the one going through xy, and (18) is the correct choice of a 1-Lipschitz function that saturates the supremum in (16).
Plugging (12) and (18) into (16) and (17), we arrive at From now on we will refer to κ xy as given in (19) as the Ricci curvature on a graph-with the understanding that the graph is either a tree, or a graph whose loops are sufficiently large to make the calculation leading to (19) valid. We will describe the latter sort of graph as "almost a tree," keeping in mind that this apparently imprecise phrase can be rendered x y x 1 x 2 x 3 x 4 y 1 y 2 y 3 Figure 3: Part of a graph which may qualify as "almost a tree." The important criterion is that the alternate route from x 1 to y 1 , passing through the top four edges, must be longer than the path from x 1 to y 1 through the edge xy.
meaningful, for instance by imposing the previously mentioned condition that loops have to have at least seven edges, with lengths varying by no more than a factor of 4/3.

Negative Ricci curvature
Consider now the Ricci curvature of the Bruhat-Tits tree with coordination number q + 1, where q = p n and we set the length of all the edges equal to a common value a. The lapse factor D x must be the same at each vertex, since in general we think of D x as a function of the edge lengths a xy . Let D be the common value of all the D x . From (19) we have which we understand as constant negative curvature. If we choose D x = d J (x), then D = (q + 1)/a 2 , and we obtain the simple result There is a peculiar feature of (21) which at first seems unattractive: the overall scale a is undetermined. In other words, we can scale the length of all vertices by a uniform factor, and we still have a graph with the same constant negative Ricci curvature. We will call this feature scale freedom. It is connected to a good feature, namely that in the linearized theory we obtain a massless equation j xy = 0 for fluctuations of bond strengths around a constant J solution. Explicitly, with the choice D x = d J (x), if we set J xy = 1 + j xy , then from (19) we find Thus if we impose (21)

A variational principle
While it is good to see a reasonable linearized equation of motion emerge from imposing constant negative Ricci curvature as in (21), we are not convinced that this is quite the optimal route to a graph theoretic version of Einstein's equations for edge length fluctuations.
The reason is that it is not clear to us how to conveniently package (21) as the variation of an action. Therefore, we would like to consider the action which appears to be at least in the spirit of the Einstein-Hilbert action with a cosmological constant Λ. Summing over all edges is similar to taking the trace of the Ricci tensor and then integrating over all of space. As before, we choose D x = d J (x), with the result that S as a whole is invariant under uniformly rescaling the lengths of all edges.
The ordinary Einstein-Hilbert action is not quite a satisfactory starting point for a variational principle, because it involves second derivatives of the metric, whereas generically to get a second-order equation of motion one wants a lagrangian density which is first order in derivatives. The well-known solution is the Gibbons-Hawking boundary term, whose effect is to cancel out the second derivative terms in the bulk Einstein-Hilbert action. We can prescribe any (smooth) region of spacetime, add the Gibbons-Hawking term on its boundary to the Einstein-Hilbert action on its interior, and derive the Einstein equations by varying the metric inside the region while holding it fixed outside. We would like to seek a similar 7 A loophole in this argument is that one could perhaps arrange for the linearized equation of motion to be j xy = 0, but to have terms at higher order in the fluctuations j xy break scale freedom. augmentation of the action (23). That is, we would like to be able to start from a large graph G, which is either a tree or "almost a tree," isolate a subgraph Σ ⊂ G, and add to the action in (23) a term on the boundary of Σ, after which we can vary the combined action on the interior of Σ and recover a second order equation of motion. Second order now means that the equation of motion should involve edges which are separated by up to two steps.
The discrete Laplace equation j xy = 0 is second order because it involves j xx i , j xy , and j yy i , and the xx i edges are two steps away from the yy i edges.
In order to realize the ideas of the previous paragraph concretely, we are going to put some restrictions on Σ, which we think of as a list of vertices and edges, where an edge is in Σ iff both the vertices of that edge are in Σ. First we require that Σ must be a finite connected subgraph of G. Consider a vertex x ∈ Σ such that at least one edge connected to x is not in Σ. There must be some such vertices, because Σ is not the whole of G, and we assume that G is connected. Let the collection of them be called ∂Σ. A crucial requirement on Σ is that for each vertex x ∈ ∂Σ, there is only one neighboring vertex, call it x , which is in Σ, and this neighboring vertex x cannot be in ∂Σ. We describe a subgraph Σ that satisfies all the restrictions we have stipulated in this paragraph as a "fat" subgraph of G, and intuitively it is like a smooth finite subregion of a manifold. Going from x ∈ ∂Σ to x is It is easy to construct the subgraphs Σ of a tree G by an iterative process: starting at a vertex x that is stipulated to be in the interior of Σ, we add all its neighboring vertices, and then additional vertices with the rule that once an additional vertex is included in Σ, we must either also add all its neighboring vertices not previously included in Σ in an earlier step, or else none of them. Of course, we must terminate this process after a finite number of steps in order to have a finite connected graph. If G has loops, then we have to be a little more careful in the choice of Σ to make sure that x is uniquely defined for every x ∈ ∂Σ. In order to be sure to have a good variational principle on all of G, we demand that G should coincide with the union of a sequence of fat subgraphs of G, each of which is a subgraph of the next.
To formulate the boundary term that we need, it is convenient first to re-express (19) as where we define a "directed half" of the Ricci curvature as and As usual we have chosen D x = d J (x). If x ∈ ∂Σ, then let's define where K 0 is some constant. Note that d J (x) and c J (x) depend on the link variables J xy on all the edges adjoining the vertex x ∈ ∂Σ, not just the edge xx belonging properly to Σ.
Likewise, κ xx refers to all these link variables. In formulating a boundary action in terms of k x and κ xx , we are going to regard J xx as dynamical (i.e. a quantity that we can vary), while the other J xy -the ones just "outside" Σ-are known but fixed.
Now we are ready to give the action for a fat subgraph Σ of a graph G which is a tree or "almost a tree:" To demonstrate that this action gives rise to a well-defined equation of motion (meaning, an equation of motion which doesn't change its form on any edge when we make Σ bigger), it is convenient first to note that we can re-express (28) as Varying S interior is straightforward: Thus if we define then the equations of motion following from the action S Σ are Clearly, a regular tree, or any regular "almost tree," with all a xy set equal to a common value a, gives a solution to the equations of motion (36). If we perturb slightly around the regular tree with a = 1 by setting J xy = 1 + j xy for all edges, then one has immediately where h µν is the induced metric on ∂Σ, and θ is the trace of the extrinsic curvature. From (38) one obtains the equation of motion R µν = − 2 2 g µν . Thus R = − 6 2 , and the bulk lagrangian is R − 2Λ = 4Λ on shell. To arrange an analogous situation in the action (28), we focus on the regular tree with coordination number q + 1 and set so that the "bulk lagrangian" κ xy − 2Λ = 4Λ when the edge length is constant. Next we inquire what value of K 0 will lead to a finite limit for S Σ as Σ grows. We choose Σ to comprise all vertices within N steps of a specified vertex C, so that ∂Σ is the set of vertices which are exactly N steps away from C. There are n v = (q + 1)q N −1 vertices in ∂Σ, and there are edges in Σ (including the ones which end on a vertex in ∂Σ). Referring to (28), we have where all the k x are assumed to have a common value k. In order to get a finite limit for S Σ as N becomes large, we must have Combining (27) and (42) we find It is easy to show that after imposing (43), S Σ has a finite limit as N → ∞. The choice (43) cancels at least the leading q N divergence in a more general circumstance, where the graph G under consideration is asymptotic to a regular tree with coordination number q + 1 and constant edge length, provided we fix the cosmological constant as in (39).

Correlators
Let's start with a total action up to boundary terms, where κ xy is defined as in (19) with our usual choice, D x = d J (x).
From this action we would like to calculate the simplest holographic correlators of an operator O dual to φ and an operator T dual to fluctuations of the bond strengths J. We will focus on correlators on the Bruhat-Tits tree T q , whose boundary is the unramified extension Q q of Q p , where q = p n . Our background "metric" consists of setting all J xy = 1. We also set all φ x = 0. The background is trivially a solution of the equations of motion following from (44). The correlators we are interested in are T T , T OO , and T T T . (The two-point function OO was computed already in [1,2].) We will work strictly at tree level in the bulk. We omit an overall prefactor multiplying S. If such a factor were included, it would simply multiply all our correlators as a prefactor.
As a convenient parametrization, we set for all edges. We make (45) the defining relation for j xy , so that it is exact rather than a linearization. To get at T T , all we need is the part of (44) quadratic in the j xy . This

Propagators
We will need the distance function d(e 1 , e 2 ) between two edges on the graph T q . By definition, d(e 1 , e 2 ) is the number of vertices one must cross in order to get from e 1 to e 2 . Similarly, the distance d(x 1 , x 2 ) between two vertices on T q is the number of edges we have to cross in order to get from x 1 to x 2 . We do not account for variable edge lengths because we are perturbing around the configuration with all J xy = 1; thus the distance function d can be thought of as characterizing the background metric.
Although our main purpose is to understand the consequences of the curvature action, we will take our calculations as far as we can with a more general action for link variables j e that includes a mass term: where the prefactor η is at this stage arbitrary. If we expand the first term of (44) to quadratic order in the fluctuations j e , the quadratic term agrees precisely with (46) provided we choose Thus we can proceed with general η and ∆ J , and at the last step specialize to massless edge length fluctuations by using (47).
Starting from the action (46), we easily see that the bulk-to-bulk Green's function for fluctuations of j e should satisfy where δ ef = 1 if e = f and 0 otherwise. One may check by direct calculation that solves (48), provided ∆ J satisfies Here and below, we use the local zeta function We will also need a bulk-to-boundary propagator, K J (e, y), where y ∈ Q q . Consider the semi-infinite path [e : y), where the notation [e indicates that e is included in the path, whereas the notation y) indicates that y is not. Let x be the vertex at the end of e that is further from y, and recall from [1] that we can identify x as a equivalence class of points (z, z 0 ), where z ∈ Q q and z 0 = p ω for some ω ∈ Z. The equivalence relation is that we regard (z, z 0 ) and (z , z 0 ) as the same point iff z = z + z 0 n for some n ∈ Z q . Then we have whereK J (e, y) = In (53), |z 0 | = p −ω is the p-adic norm of z 0 , and the norm in the denominator is |(z 0 , y −z)| ≡ sup{|z 0 |, |y − z|}. By construction, K J (e, y) satisfies the bulk equation and its integral over the boundary is Finally, K J satisfies the property where f is any edge along the path [e : y). In section 4.2 we will need a Fourier integral of where e is an edge on the path in T q from ∞ to 0, and z 0 is the depth coordinate of the vertex of e further from the boundary point 0. In (57), χ(ξ) is an additive character on Q q with the property χ(ξ) = e 2πiξ for rational ξ (see for example [1] for details on the Fourier transform over Q q ). The function γ(ξ) is 1 when ξ ∈ Z q , and 0 otherwise.

Two-point function
To compute the two-point function T (z 1 )T (z 2 ) for separated points z 1 , z 2 ∈ Q q , we must evaluate the quadratic on-shell action (46) on a solution to the equations of motion. For a solution to the equation of motion, (46) reduces to Because we are interested in separated points, we will not attempt to track boundary terms as we did for the curvature action in section 3.2.
We employ the familiar Fourier space method, where we label each edge e by coordinates (z 0 , z), where z 0 = p ω for some ω ∈ Z and z ∈ Q q . The meaning of this labeling is that the vertices at the ends of the edge e are associated to (z 0 , z) and (pz 0 , z), where z 0 = p ω for some ω ∈ Z, and z ∈ Q q is defined up to replacements z → z + pz 0 n for n ∈ Z q . Guided by (57), we set where we define 8 and In (59)-(60), we have introduced a UV cutoff = p Ω , and we prescribe a cutoff form of the on-shell action (58) as follows: Each sum in square brackets is over all edges with a fixed z 0 , as indicated. Plugging (59) into (62), we obtain a regulated two-point function The non-universal terms include divergent terms with no dependence on k 1 and k 2 other than δ(k 1 + k 2 ), and also terms that are subleading relative to the term shown in the last line of (63) in the limit where |k 1 | and |k 2 | are small. Referring to [1], we have up to divergent terms proportional to δ(x). Thus, for separated points, we find where we have attached a leg factor for the operator T (z):

The mixed three-point function
To compute the mixed three-point function T (z 1 )O(z 2 )O(z 3 ) for separated points z 1 , z 2 , and z 3 , we require the cubic interaction term that follows from the second term in (44): In addition to the bulk-to-boundary propagator (52) for edge fluctuations, we need the bulkto-boundary propagator for φ x , known from [1]: where now (z 0 , z) is understood to be a coordinate choice for the bulk vertex a.
The three-point function can be calculated as follows: and thereforeK The quantityÂ T OO in (72) has no dependence on the z i and comes from the summation over all edges in (68). Explicit calculation ofÂ T OO is unilluminating, and we will quote here only the result: A significant simplification occurs when we take ∆ J → n, as appropriate for massless edge length fluctuations: then the three-point function becomes (76)

The purely geometric three-point function
To compute the three-point function T (z 1 )T (z 2 )T (z 3 ) for separated points, we only need the first term in (44). Expanding this curvature action to cubic order in the fluctuations j xy , we obtain the interaction terms where where as usual x i denotes the vertices adjacent to x other than y, while y i denotes the vertices adjacent to y other than x. Similarly to (71)-(72), we can easily see that where C is the vertex in T q where paths from z 1 , z 2 , and z 3 meet, and each C i is the vertex next to C one step closer to the corresponding z i . The factorÂ T T T has no dependence on the z i , and for generic values of the coefficients c i it is non-vanishing; however, remarkably, for the particular values (78), we findÂ T T T = 0. (This is for ∆ J = n; in contrast to previous subsections, we do not consider general ∆ J here.) Consider first the contribution of the j 3 xy interaction in (77) to the three-point function: It is where In (81) we have introduced functions on the tree. By construction, h i (e) increases by a factor of p ∆ J for each step that e takes along the path from C to z i in the direction of z i . But it decreases by a factor of p −∆ J for each step that e takes off of this path. Intuitively, h i (e) is like the bulk-to-bulk propagator G( CC i , e), but when the path from CC i to e has vertices in common with the path from C to z i , h i (e) includes extra positive powers of p ∆ J (relative toĜ( CC i , e)) to account for back-tracking.
Following steps similar to (80) for the remaining terms in (77), we find where In (84) we have summations over all permutations σ in the symmetric group S 3 . The reason is that we must be able to map any permutation of the three edges CC i to the three edges involved in the interactions (77). A similar sum implicitly entered into (81), but it gave only the prefactor of 6 because the interaction term j 3 xy doesn't distinguish among the different permutations. The end result of performing the sums in (81) and (84) is and plugging into (83) results inÂ T T T = 0 upon using the coefficients (78). Thus the three-point function vanishes: for separated points z 1 , z 2 , and z 3 . It may be noted that (79) does not account for boundary terms in the action. Because such terms (at least, the boundary terms we found in section 3.2) are local on the boundary, they do not affect the result (86) for separated points. A proper understanding of contact terms undoubtedly does require an account of boundary terms.

Solutions to the discrete Einstein equations
We saw in section 3 (equations (34)-(36) in particular) that the discrete version of the Einstein equation takes the form γ x→y + γ y→x = 0, where γ x→y is a "directed half" of the variation of the edge length action with respect to J xy . The only solutions we have exhibited so far are the trivial ones where J xy is constant for all edges, and these solutions trivially satisfy the stronger equations γ x→y = 0, which can be recast as Clearly, setting all the J xy to a common value solves (87) on any graph G, regular or not, with or without loops. Perhaps less obviously, constant J xy is the only solution to (87), provided only that G is connected. To see this, let x be a fixed vertex, and sum (87) over all y adjacent to x. We get where q x +1 indicates the coordination number of the vertex x (the number of edges connected to it), and c J (x) = y∼x J xy while d J (x) = y∼x J xy as in previous sections. Now define two vectors in R qx+1 : Here and below, we use x 1 , x 2 , . . . , x qx+1 to denote the neighboring vertices of a given vertex x. It is illuminating to rewrite (88) as Recalling that the Cauchy-Schwarz inequality, ( v · b) 2 ≤ v 2 b 2 , is saturated only when v and b are linearly dependent, we see that all entries in b must in fact be identical. Replaying the argument for each vertex x in G, we see that the edge lengths around each vertex must be equal, and that means that a xy is the same for all edges in G given that it is a connected graph.
We are now going to write a more explicit form of the discrete Einstein equations (36) which will make it easier to find solutions with non-constant edge lengths. In the discussion to follow, the graph G can still be a general connected graph. However, the discrete Einstein equations are well motivated (at least, according to our development) only when G is "almost a tree" in the sense explained in section 3. To proceed, we introduce the positive quantities and we observe that the discrete Einstein equations can be rewritten in the form whose general solution is parametrized by an angular variable θ xy ∈ (−π/4, 3π/4) (see figure 5): The form (93) refers implicitly to a direction on the edge xy, in that λ x→y is expressed in terms of cos θ xy while λ y→x is expressed in terms of sin θ xy . To make the notation more symmetrical, let's introduce θ x→y = θ xy and θ y→x = π 2 − θ xy . Also introduce for all neighboring x and y. Then (93) reduces to Plugging (91) into (95) and rearranging, we wind up with (To see this, it helps to note that ρ 2 = qσ 2 + 2σ − 1.) If we think of J xx k as fixed, then (96) has an obvious geometrical interpretation. Consider the space R qx with coordinates ( J xx 1 , . . . , J xx k , . . . , J xx qx+1 ), meaning all the To recover the constant J xy solutions from (96), we set all θ x→y = π/4, so that σ x→y = 1/2, ρ 2 x→y = q x /4, and (96) is trivially satisfied for all x and all neighboring x k . We would now like to exhibit a non-trivial solution on a graph with the topology of T q for odd q, based on the idea that half the bond strengths leading into a given vertex take one value, while the other half take a different value. (It doesn't matter whether q = p n for some odd prime p.) Pick a particular angle α ∈ (−π/4, 3π/4), not equal to π/4, and setα = π 2 − α. Let us abbreviate notation by setting σ = σ(α),σ = σ(α), ρ = ρ(α), andρ = ρ(α). Then at each vertex x, we set where the J x are as yet undetermined real positive numbers. Already, (97) passes a nontrivial test: namely, (96) is satisfied both for odd and even k, due to the unobvious but easily What remains is to check that the vertices can be tied together so that the assignments (97) are consistent when applied to all vertices. Let y be one of the neighbors of x, so that y = x k for some k. It must be that x = y for some , where the y i are all the neighbors of y.
The edge xx k is also the edge yy , and we can look at consistency conditions on this edge.
The assignments of θ x→x i in (97) immediately lead us to conclude that k and must have opposite parity. This is because if θ x→y = α, then θ y→x =α by definition of θ x→y and θ y→x . Now that we have a consistent assignment of θ x→y and θ y→x , we can ask about the bond strength between x and y. Assume k is even. Then J xx k =σ 2 J x from the assignments at vertex x, while J yy = σ 2 J y from the assignments at vertex y. But the edges xx k and yy coincide: they are both the edge xy. Thus we see that J y = (σ/σ) 2 J x . If instead k is odd, then the same reasoning would lead us to J y = (σ/σ) 2 J x . Continuing, we see that if a vertex z can be reached from a fixed vertex x along a path where N even of the directed links have the form ww i with i even, while N odd have the same form with i odd, then The final configuration of bond strengths is unique up to relabeling of vertices and an overall rescaling of all the J xy . See figure 6. We note that the solution we have exhibited is very different from constant J xy , in that the variation in the J xy is exponential with respect to the number of steps along the graph. As a result, many paths to the boundary have finite distance, while others have an exponentially diverging distance, and still others have the linearly diverging distance that one encounters in constant J xy solutions. If a distance function can be induced on the boundary through some procedure of regulation starting from Figure 6: A regular tree (for q = 3) with non-constant edge lengths as described by (97) where we recognize the q dependent part to be the Ricci curvature of a constant edge solution, computed in (21). The Ricci curvature given in (100) displays scale freedom just like the constant edge solution, and it is negative for all q ≥ 3 with −π/4 < α < 3π/4. The non-constancy of the edges simply makes the Ricci curvature less negative compared to the constant edge solution.
Analogous to the construction of the BTZ black hole, we can quotient the non-uniform tree by certain abelian subgroups of the isometry group of the tree. The resulting geometry is "almost a tree" with precisely one cycle consisting of an even number of links. The edge lengths along the cycle are not necessarily all the same; different configurations are possible from the same non-uniform tree, depending on different choices of the abelian subgroup. We leave the detailed study of such topologies for future work.

Conclusions
Using the ideas of [7,8], we have formulated an action principle for edge length dynamics on a graph in terms of Ricci curvature. The action (28) is a discrete version of the Einstein-Hilbert action with a cosmological constant and a Gibbons-Hawking boundary term, and it has a well-defined variational principle leading to discrete Einstein equations (36). In contrast to many lattice constructions, there is no intention of taking a continuum limit, at least when we have p-adic AdS/CFT in view. The Bruhat-Tits tree T p , which stands in for anti-de Sitter space in p-adic AdS/CFT, is naturally discrete, and the obvious p-adic conformal symmetries act on the tree as graph isometries: see Appendix A.
While there are substantial similarities between edge length dynamics and Einstein gravity, there are some key differences. Most notably, in our construction, we do not get spin 2 gravitons in any obvious sense. The field theory operator T dual to edge length fluctuations on the Bruhat-Tits tree T p has a two-point function T (z)T (0) ∝ 1/|z| 2 , like a scalar operator. As discussed in [2], higher spin would be characterized by a more general multiplicative character. When we generalize to the unramified extension Q p n , which is an n-dimensional vector space over Q p , we find T (z)T (0) ∝ 1/|z| 2n , meaning that T (z) has dimension n, as expected for a stress tensor; but still there is no spin. Perhaps even more surprising, the three-point function T (z 1 )T (z 2 )T (z 3 ) vanishes for separated points, though this is a result which seems to depend rather sensitively on the precise construction of the Ricci curvature; in particular, it depends on our choice of the lapse factor D x to be the sum d J (x) of the bond strengths for edges adjoining the vertex x. There are many directions to go from here. The action (28) seems ideally suited for an analysis of the free energy of graphs such as the non-archimedean black holes of [3,2].
The results of section 4 on correlators invite an analysis in p-adic field theory of what we should mean by a stress energy tensor. While p-adic applications obviously privilege regular graphs with at most finitely many cycles, we can investigate a much broader class of graphs. For example, tessellations of the Poincaré disk could be considered, provided all cycles are sufficiently long. Perhaps some connection between our edge length dynamics and entanglement constructions along the lines of [2,13,14] could be made explicit. We look forward to reporting on these topics in the future.

Acknowledgments
The work of S. Gubser

Appendices
A GL 2 transformations of edges and vertices in a uniform tree Here we review some discussion of [15,16] about how subgroups of GL 2 (Q p ) acts on edges and vertices of the tree. For a tree of constant negative curvature with uniform edge lengths, these GL 2 properties continue to hold. Much like in classifications of spin representations of the Lorentz group, we can perform a translation so that a given vertex is moved to the origin, then consider transformations that leave the origin fixed. This will tell us a bit about how fields at vertices like φ(x) and fields on edges like J xy behave under such transformations.
Recall that the nodes of the Bruhat-Tits tree are lattices in Q 2 p modulo similarity transformations. If u and v form a basis of Q 2 p , call the lattice they span [u, v]. If g is in GL 2 (Q p ), acting with g on the lattice takes us to another lattice [gu, gv]. So GL 2 moves vertices around in the tree, and it turns out to also preserve edges of which there are p + 1 per vertex. A convenient origin x 0 of the Bruhat-Tits tree is defined by the lattice The total Bruhat-Tits tree with origin x 0 is the coset PGL 2 (Q p )/PGL 2 (Z p ) (we've used P to take care of the similarity.) PGL 2 (Z p ) is the maximal compact subgroup and thus fixes the origin x 0 . One can see that the origin is fixed by this stabilizer by explicit matrix multiplication of the basis vectors with Z p coefficients; the resulting lattice will always be Z 2 p up to similarity. The nodes 1 step from x 0 are labeled by elements of P 1 (F p ). This is the set of nonzero pairs (z 1 , z 2 ) in Z/pZ modulo scalar multiplication in this group. Explicitly these are the lattices [pu 0 , v 0 ] and [u 0 + nv 0 , pv 0 ] for n = 0, . . . , p − 1. These adjacent vertices x ∼ x 0 are permuted by the action of SL(2, Z p ). This is analogous to the SO(2) ⊂ SL 2 (R) action on the upper half plane.
Given a local field φ(x) living at a vertex in the tree, we are free to make a GL 2 transformation to translate this field to the origin, φ(x 0 ). Further SL(2, Z p ) transformations leave this invariant, and φ would appear to have the expected scalar character under the stabilizer group. For a generic field living on an edge U xy , we can again perform a GL 2 transformation to map this to U x 0 x . As should be clear from the geometry, for a Λ ∈ SL(2, Z p ), the x index will transform as U x 0 x = Λ xx U x 0 x . We should not be too cavalier about calling this a spin, as in ordinary AdS different possible coordinate systems and choices of stabilizer lead to different linear combinations of AdS isometries.
We have so far discussed the maximal compact subgroup of GL 2 which fixes the origin, and we can also find a transformation which fixes a neighbor x 1 . The neighbor is obtained by applying so that α(x 0 ) = x 1 . For K a GL(2, Z p ) matrix, αKα −1 fixes x 1 . We can now look for an operation which fixes both x 0 and x 1 and by construction the edge connecting them. This is found by intersection of the two stabilizer groups; K ∩ αKα −1 fixes the oriented edge of the tree and rotates all the branches running away from each endpoint. By conjugation every edge possess such a stabilizer.
The fact that edge variables U xy transform trivially under this new set of stabilizers may make classification of spin representations more delicate. This may explain why the gravitational degrees of freedom discussed in the present work do not appear to have spin.
We leave further exploration of this idea for future work.

B Vladimirov derivatives
In this appendix, we recall various definitions of the Vladimirov derivative operator (which is a non-local operation defined on real functions of a p-adic variable), and clarify some of its properties. The Vladimirov derivative is important in the context of p-adic AdS/CFT as the derivative operator appearing in the boundary theory, for instance in the action for the p-adic free boson CFT. While none of the results stated here are new, they have not (as far as we know) been clearly and explicitly summarized in previous literature.
One commonly stated definition of the Vladimirov operator D α is where α is a real parameter representing the order of the derivative. This definition is puzzling for several reasons: most importantly, it's not obvious that it does what it's supposed to do (multiplication by |k| p ) in the Fourier domain. Furthermore, it's not clear that it has the right composition properties. We would like it to hold that As we will show, one should understand (103) as a regularized version of the other definition occurring in the literature: where the denotes convolution, and the family of kernels π α are defined by π α (x) = |x| α−1 p Γ p (α) .
Note that plugging this definition into (105) yields the first term, but only the first term, of (103). When f (x) is nonzero, the second term is in fact infinite, at least for α = 1; it diverges due to the pole in the integrand as y → x.
In fact, this is true as long as all of the expressions involved converge; this happens when α > 0, β > 0, α + β < 1. The general result then follows by analytic continuation; what this amounts to is that we have to allow ourselves to resum geometric series, even if the series fail to converge. Similar behavior will occur in our analysis of the definitions of derivative.
First of all, let's note that the class of well-behaved functions we're interested in are locally constant, and that the space of such functions is spanned by characteristic functions of p-adic open sets: for instance, Since both definitions of derivative are linear and translation-invariant, we need only check their equivalence on the functions γ ν to establish it in general.
Let's start with the definition by convolution, There are two cases to consider: firstly, when |x| p > p −ν (so that the pole is outside the support of the characteristic function and can't cause divergences), and |x| p ≤ p −ν . In the first case, |x − y| = |x|, and the integrand is just a constant over the region of integration; we obtain where the last factor comes from the measure of the set p ν · Z p .
In the second, more complicated case, there are three sub-cases to consider: |y| can be strictly less than x, greater than, or equal. We write the integral as a sum over the circles ord p y = µ; recall that the measure of each such circle is just (p − 1)/p 1+µ . Using the ultrametric property of the norm, and adopting the notation λ = ord p x, we find that