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JHEP06(2017)157

Published for SISSA by Springer

Received: May 18, 2017

Accepted: June 18, 2017

Published: June 30, 2017

Edge length dynamics on graphs with applications to

p-adic AdS/CFT

Steven S. Gubser,

Matthew Heydeman,

Christian Jepsen,

Matilde Marcolli,

Sarthak Parikh,

Ingmar Saberi,

Bogdan Stoica

e,f

and Brian Trundy

Joseph Henry Laboratories, Princeton University,

Princeton, NJ 08544, U.S.A.

Walter Burke Institute for Theoretical Physics, California Institute of Technology,

452-48, Pasadena, CA 91125, U.S.A.

Department of Mathematics, California Institute of Technology,

253-37, Pasadena, CA 91125, U.S.A.

Mathematisches Institut, Ruprecht-Karls-Universit¨at Heidelberg,

Im Neuenheimer Feld 205, 69120 Heidelberg, Germany

Martin A. Fisher School of Physics, Brandeis University,

Waltham, MA 02453, U.S.A.

Department of Physics, Brown University,

Providence RI 02912, U.S.A.

E-mail: ssgubser@princeton.edu, mheydema@caltech.edu,

cjepsen@princeton.edu, matilde@caltech.edu, sparikh@princeton.edu,

saberi@mathi.uni-heidelberg.de, bstoica@brandeis.edu,

btrundy@princeton.edu

Abstract: 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.

Keywords: Lattice Models of Gravity, AdS-CFT Correspondence, Classical Theories of

Gravity

ArXiv ePrint: 1612.09580

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP06(2017)157

JHEP06(2017)157

Here

hxyi

indicates a sum over edges (i.e. without counting hxyi and hyxi separately),

and a

is the length of the edge xy, while V is a potential for the bulk scalar ﬁeld φ

Calculations of correlators of the operator dual to φ

were carried out in [1, 2] with all a

set equal to 1, and these calculations have notable precursors in the literature on p-adic

strings, for example [4].

Now we would like to ask what interesting dynamics for the edge

lengths a

could be added.

To get started, let’s set

, (1.2)

where e = xy is an edge. Then J

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 Gaus-

sian white noise in the J

: that is, we could draw each J

independently from a Gaussian

distribution. White noise for the J

seems quite unlike gravitational dynamics, because

nearby J

don’t pull on one another. Better would be to introduce some interactions among

the J

on neighboring edges by adding to the (1.1) an action

hefi

− J

)

U(J

) , (1.3)

where hefi means a sum over neighboring edges — that is, edges which share one vertex. If

we omitted the ﬁrst term in (1.3), and made the potential U quadratic, then the J

would

be independent from one another, and we would be back to the case of Gaussian white

noise (but unquenched assuming we form a partition function Z =

DJDφ e

−S

). In

particular, we see that a quadratic term in the U corresponds to a mass term for the

edge variables J

. Probably for something resembling gravity, we should avoid having a

quadratic term in the U.

While (1.3) 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 [6–8]. 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

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.

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

on vertices

and introducing θ-φ interactions, we don’t think of it as such an interesting avenue.

– 2 –

JHEP06(2017)157

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 (1.3) 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

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.

2 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.

2.1 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

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

– 3 –

JHEP06(2017)157

integers, neither of which is divisible by p, then

|x|

= p

−v

when x = p

. (2.1)

By deﬁnition, |0|

= 0. We will usually drop the subscript p and write |x| instead of

|x|

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

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:

x = p

+ c

p + c

+ . . .) , (2.2)

where v ∈ Z, c

∈ F

, and c

∈ F

. Here F

denotes the non-zero elements in F

The

inﬁnite series in (2.2) appears to be highly divergent, but in fact it converges because the

are bounded in p-adic norm, while |p

| = p

−n

. The expansion (2.2) 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

, can be understood informally as a graphical

representation of the expansion (2.2). 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 (2.2)). 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 c

∈ F

. In each subsequent step, we have to choose among p possible

directions, and each such choice can be labeled by an element c

∈ F

. 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

∪ {∞}, which is P

It can be shown that the Bruhat-Tits tree is a quotient space:

PGL(2, Q

)

PGL(2, Z

)

, (2.3)

where Z

denotes the p-adic integers (the completion of Z with respect to |·|

, or equivalently

the set of all x ∈ Q

with |x|

≤ 1). The quotient (2.3) is similar to the realization of the

p-adic numbers in Q

add and multiply with carrying, so strictly speaking c

and c

take values in

{1, . . . , p − 1} and {0, . . . , p − 1} respectively, and not in F

and F

. For the sake of conciseness we will

suppress this technical detail in the rest of the paper.

If we were attempting to be rigorous, we could have started by deﬁning the set of boundary points as

the set of semi-inﬁnite paths (with no back-tracking) starting from some speciﬁed vertex C of the tree.

– 4 –

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