Relations between scaling exponents in unimodular random graphs

We investigate the validity of the"Einstein relations"in the general setting of unimodular random networks. These are equalities relating scaling exponents: $d_w = d_f + \tilde{\zeta}$ and $d_s = 2 d_f/d_w$, where $d_w$ is the walk dimension, $d_f$ is the fractal dimension, $d_s$ is the spectral dimension, and $\tilde{\zeta}$ is the resistance exponent. Roughly speaking, this relates the mean displacement and return probability of a random walker to the density and conductivity of the underlying medium. We show that if $d_f$ and $\tilde{\zeta} \geq 0$ exist, then $d_w$ and $d_s$ exist, and the aforementioned equalities hold. Moreover, our primary new estimate is the relation $d_w \geq d_f + \tilde{\zeta}$, which is established for all $\tilde{\zeta} \in \mathbb{R}$. For the uniform infinite planar triangulation (UIPT), this yields the consequence $d_w=4$ using $d_f=4$ (Angel 2003) and $\tilde{\zeta}=0$ (established here as a consequence of the Liouville Quantum Gravity theory, following Gwynne-Miller 2017 and Ding-Gwynne 2020). The conclusion $d_w=4$ had been previously established by Gwynne and Hutchcroft (2018) using more elaborate methods. A new consequence is that $d_w = d_f$ for the uniform infinite Schnyder-wood decorated triangulation, implying that the simple random walk is subdiffusive, since $d_f>2$ (Ding and Gwynne 2020). For the random walk on $\mathbb{Z}^2$ driven by conductances from an exponentiated Gaussian free field with exponent $\gamma>0$, one has $d_f = d_f(\gamma)$ and $\tilde{\zeta}=0$ (Biskup, Ding, and Goswami 2020). This yields $d_s=2$ and $d_w = d_f$, confirming two predictions of those authors.

where is the walk dimension, is the fractal dimension, is the spectral dimension, and ˜ is the resistance exponent.Roughly speaking, this relates the mean displacement and return probability of a random walker to the density and conductivity of the underlying medium.We show that if and ˜ 0 exist, then and exist, and the aforementioned equalities hold.Moreover, our primary new estimate + ˜ , is established for all ˜ ∈ ℝ.
For the uniform infinite planar triangulation (UIPT), this yields the consequence = 4 using = 4 (Angel 2003) and ˜ = 0 (established here as a consequence of the Liouville Quantum Gravity theory, following Gwynne-Miller 2020 and Ding-Gwynne 2020).The conclusion = 4 had been previously established by Gwynne and Hutchcroft (2018) using more elaborate methods.A new consequence is that = for the uniform infinite Schnyder-wood decorated triangulation, implying that the simple random walk is subdiffusive, since > 2.
For the random walk on ℤ 2 driven by conductances from an exponentiated Gaussian free field with exponent > 0, one has = ( ) and ˜ = 0 (Biskup, Ding, and Goswami 2020).Thus the scaling relations yield = 2 and = , confirming two predictions of those authors.

Introduction
Consider an infinite, locally-finite graph and a subgraph of .For ∈ ( ), let ( , ), denote the graph ball of radius , and let ˜ ( , ) := ( , ) ∩ ( ) denote this ball restricted to .Let ( , ) denote the path distance between a pair , ∈ ( ).Denote by { } the simple random walk on , and the discrete-time heat kernel We write R eff ( ↔ ) for the effective resistance between two subsets , ⊆ ( ).
For a variety of models arising in statistical physics, certain asymptotic geometric and spectral properties of the graph are known or conjectured to have scaling exponents: where one takes , → ∞, but we leave the meaning of "∼" imprecise for a moment.These exponents are, respectively, referred to as the fractal dimension, walk dimension, spectral dimension, and resistance exponent.We refer to the extensive discussion in [BH00, Ch. 5-6].
Moreover, by modeling the subgraph as a homogenous underlying substrate with density and conductivity prescribed by and ˜ , one obtains the plausible relations (1.3) In the regime ˜ > 0, these relations have been rigorously verified under somewhat stronger assumptions in the setting of strongly recurrent graphs (see [Tel90,Tel95]) and [Bar98,KM08,Kum14b]).In the latter set of works, the most significant departure from our assumptions is the stronger requirement for uniform control on pointwise effective resistances of the form max R eff ( ↔ ) : ∈ ( , ) ˜ + (1) , ∈ ( ). (1.4) Such methods have been extended to the setting where ( , ) is a random rooted graph ( [KM08,BJKS08]) under the statistical assumption that these relations hold sufficiently often for all sufficiently large scales, and only for balls around the root.
Our main contribution is to establish (1.2) and (1.3) under somewhat less restrictive conditions, but using an additional feature of many such models: Unimodularity of the random rooted graph ( , ).When ˜ 0, it has been significantly more challenging to characterize situations where (1.2)-(1.3)hold; see, for instance, Open Problem III in [Kum14a].Our main new estimate is the speed relation + ˜ , which is established for all ˜ ∈ ℝ.In particular, this shows that the random walk is subdiffusive whenever + ˜ > 2, and applies equally well to models where the random walk is transient.Let us now highlight some notable settings in which the relations can be applied.
The IIC in high dimensions.As a prominent example, consider the resolution by Kozma and Nachmias [KN09] of the Alexander-Orbach conjecture for the incipient infinite cluster (IIC) of critical percolation on ℤ , for sufficiently large.If ( , 0) denotes the IIC, then in our language, = , as they consider the intrinsic graph metric, and establish that for every > 1 and 1, with probability at least 1 − ( ), it holds that where ( ) ( − ) for some > 1.One should consider this a statistical verification that = 2 and ˜ = 1, as in this setting, one gets the analog of (1.4) for free free from the trivial bound The uniform infinite planar triangulation.Consider, on the other hand, the uniform infinite planar triangulation (UIPT) considered as a random rooted graph ( , ).In this case, Angel [Ang03] established that almost surely and Gwynne and Miller [GM21] showed that almost surely lim This falls short of verifying (1.1).Nevertheless, we show in Section 4.3 that ˜ = 0 is a consequence of the Liouville Quantum Gravity (LQG) estimates derived in [DMS14, GM21, GMS19, GHS20, DG20].But while the known statistics of | ( , )| are suitable to allow application of the strongly recurrent theory, this does not hold for the effective resistance bounds.This is highlighted by Gwynne and Hutchcroft [GH20] who establish = 4 using even finer aspects of the LQG theory.The authors state "while it may be possible in principle to prove 4 using electrical techniques, doing so appears to require matching upper and lower bounds for effective resistances [...] differing by at most a constant order multiplicative factor."Our methods show that, when leveraging unimodularity, even coarse estimates with subpolynomial errors suffice.
It is open whether ˜ = 0 or = 4 for the uniform infinite planar quadrangulation (UIPQ), but our verification of (1.2) shows that only one such equality needs to be established.

Random planar maps in the -LQG universality class.
More generally, we will establish in Section 4.3 that ˜ = 0 whenever a random planar map ( , ) can be coupled to a -mated-CRT map with ∈ (0, 2).The connection between such maps and LQG was established in [DMS14].
This family includes the UIPT (where = 8/3).Ding and Gwynne [DG20] have shown that exists for such maps, and Gwynne and Huthcroft [GH20] established that = for most known examples, but not for the uniform infinite Schnyder-wood decorated triangulation [LSW17] (where = 1), for a technical reason underlying the construction of a certain coupling (see [GH20, Rem.2.11]).We mention this primarily to emphasize the utility of a general theorem, since it is likely the technical obstacle could have been circumvented with sufficient effort.
In this case, one has (See below for the definition of when the edges have conductances.)In Section 4.4, we recall the model formally and observe that the paper [BDG20] contains estimates that establish ˜ = 0 for every > 0. Hence the relations (1.2)-(1.3)yield = and = 2, both of which were conjectured in [BDG20], though only annealed estimates were obtained.(See Section 1.3.1 for a brief discussion of why our approach yields two-sided quenched bounds.) The IIC in dimension two.Consider the incipient infinite cluster for 2D critical percolation [Kes86], which can be realized as a unimodular random subgraph ( , 0) of = ℤ 2 [J 03].It is known that = 91/48 in the 2D hexagonal lattice [LSW02,Smi01], and the same value is conjectured to hold for all 2D lattices regardless of the local structure.
Existence of the exponent ˜ is open for any lattice; experiments give the estimate ˜ = 0.9825 ± 0.0008 [Gra99].The most precise experimental estimate for = 2.8784 ± 0.0008 is derived from estimates for ˜ , and our verfication of (1.2) puts this on rigorous footing (assuming, of course, that ˜ is well-defined).Indeed, one motivation for our work was the question of whether the exponent should be a conformal invariant of critical 2D percolation, and it is plausibly more tractable to establish this for ˜ .
Throughout, we will make the following mild boundedness assumption: This is analogous to the assumption [deg ( )] < ∞ that appears often in the setting of unimodular random graphs, which are defined in Section 2.3 when we need to employ the Mass-Transport Principle.
For now, it suffices to say that there is a one-to-one correspondence: Indeed, if and ˜ are the respective measures, then the correspondence is given by a change of law .
where / ˜ is the Radon-Nikodym derivative.We refer to [AL07] for an extensive reference on unimodular random graphs, and to [BC12, Prop.2.5] for the connection between unimodular and reversible random graphs.

Almost sure scaling exponents
and define the walk exponents and by assuming the corresponding limits exist.In that case we, we will use the language " exists" or " Let us define upper and lower resistance exponents.Define ˜ and ˜ 0 as the largest and smallest values, respectively, such that, for every > 0, almost surely, for all but finitely many ∈ ℕ: where we have denoted the complement of ( , ) in by The exponents ˜ ˜ 0 always exist and ˜ 0 0. The exponent ˜ is referred to as the "resistance exponent" in the statistical physics literature (see [BH00, §5.3]); see Remark 1.2 below.We emphasize that all the exponents we define are not random variables, but functions of the law of ( , ).Our main theorem can then be stated as follows.
In the next section, we control the annealed variants as well, where one takes expectations over the random walk.
Theorem 1.1.Suppose that ( , ) is a reversible random network satisfying [1/ ] < ∞.If exists and ˜ = ˜ 0 , then the exponents , , and exist and it holds that See Corollary 1.6 for further equalities involving annealed versions of and .
Remark 1.2 (The resistance exponents).The resistance exponent is usually characterized heuristically as the value ˜ such (1.10) So the left-hand side of (1.9) would naturally be replaced by The lower bound we require is substantially weaker, allowing one to consider spatial fluctuations of magnitude (1) .The upper bound in (1.9), on the other hand, is somewhat stronger than (1.10), and encodes a level of spectral regularity.For instance, if satisfies an elliptic Harnack inequality and is "strongly recurrent" in the sense of [Tel06, Def.2.1], then See [Tel06, Thm.4.6] and Theorem 4.7.
Comparison to the strongly recurrent theory.Let us try to interpret the strongly recurrent theory (cf.Assumption 1.2 in [KM08]) in the setting of subpolynomial errors.The resistance assumptions would take the form: For every > 0, almost surely, for sufficiently large: (1.12) These assumptions imply that when > 0, it holds that ˜ = ˜ 0 = ; this is proved in Theorem 4.7.
Hence the theory we present (in the setting of unimodular random graphs) is more general, at least in terms of concluding the relations (1.2) and (1.3).Under assumptions (1.11) and (1.12), one can uniformly lower bound the Green kernel g ( , ) ( , ) (see Section 4.2 for definitions) for all points ∈ ( , ) and some .In other words, every point in ( , ) is visited often on average before the random walk exits ( , ).See, for instance, [BCK05, §3.2].This yields a subdiffusive estimate on the speed of the random walk, specifically an almost sure lower bound on [ | ( , )].
Instead of a pointwise bound, we use a lower bound on ˜ to deform the graph metric (see the next section).The effective resistance across an annulus being large is equivalent to its discrete extremal length being large (see Section 2.1).Thus in most scales and localities, we can extract a metric that locally "stretches" the space.By randomly covering the space with annuli at all scales, we obtain a "quasisymmetric" deformation (only in an asymptotic, statistical sense) that is bigger by a power than the graph metric.This argument is similar in spirit to one of Keith and Laakso [KL04, Thm.5.0.10] which shows that the Assouad dimension of a metric measure space can be reduced through a quasisymmetric homeomorphism if the discrete modulus across annuli is large.
Finally, by applying Markov type theory, we bound the speed of the walk in the stretched metric, which leads to a stronger bound in the graph metric.

Upper and lower exponents
Even when scaling exponents do not exist, our arguments give inequalities between various superior and inferior limits.Given a sequence {ℰ : 1} of events on some probability space, let us say that they occur almost surely eventually (a.s.e.) if ℙ[#{ 1 : ¬ℰ } < ∞] = 1.For a family { } of random variables, we will define ¯ and ¯ to be the largest and smallest values, respectively, such that for every > 0, almost surely eventually, where we allow the exponents to take values {−∞, +∞} if no such number exists.Note that ∼ ∼ ∼ (i.e., the exponent "exists") if and only if ¯ = ¯ .Let us consider the corresponding extremal exponents such that for every > 0 the following relations hold almost surely eventually (with respect to , 1): We will establish the following chains of inequalities, which together prove Theorem 1.1. and To see that this yields Theorem 1.1, simply note that when ˜ = ˜ 0 and ¯ = ¯ , then the upper and lower bounds in (1.18) and (1.13) match, and the upper and lower bounds in (1.19) are both equal to 2 / because the first set of inequalities implies = + ˜ .
Remark 1.4 (Negative resistance exponent).For ˜ < 0, Theorem 1.3 yields (assuming , exist): Without further assumptions, the final inequality cannot be made tight.Indeed, for every > 0, there are unimodular random planar graphs of almost sure uniform polynomial growth and ˜ −1 + [EL20].Yet these graphs must satisfy 2 [Lee17].In the general setting of Dirichlet forms on metric measure spaces, the "resistance conjecture" [GHL15, pg.1493] asserts conditions under which (1.2)-(1.3)might hold even for ˜ < 0. The primary additional condition is a Poincaré inequality with matching exponent.In our setting, the existence of does not yield the "bounded covering" property, that almost surely every ball ( , ) can be covered by (1) balls of radius /2.It seems likely that a variant of this condition should also be imposed to recover (1.2)-(1.3).
Let us give a brief outline of how Theorem 1.3 is proved.The unlabeled inequality is trivial.Both inequalities (1.14) and (1.17) are a straightforward consequence of Markov's inequality and the Borel-Cantelli lemma.The content of inequalities (1.15) and (1.16) lies in the relations ¯ ¯ and ¯ ¯ .These follow from the elementary inequality Let us remark that, when ( , ) is a reversible random network, then ( , , ) is also a reversible random network if is independent of conditioned on the isomorphism class of .Our constructions will have this property.The next theorem (proved in Section 3) is a variant of the approach pursued in [Lee17].
Strictly speaking, since we allow to take the value 0, this is only a pseudometric, but that will not present any difficulty.If the effective resistance across is at least ˜ , then by the duality between effective resistance and discrete extremal length (see Section 2.1), there is a length functional : where [ ] is the subgraph induced on .Let us suppose that the total volume in satisfies and we normalize to have expectation squared 1 under the measure ({ , })/ on ( [ ]): . This yields: dist meaning that, with normalized unit area, ˆ "stretches" the graph annulus by a positive power when ˜ + > 2 (see Figure 1(a)).If is sufficiently regular (e.g., a lattice), then we could tile annuli at this scale (as in Figure 1(b)) so that if we define as the sum of the length functionals over the tiled annuli, then for any pair , ∈ ( ) with ( , ) 1+ and at least one of or in the center of an annulus, we would have dist ( , ) ( ˜ + )/2 .In a finite-dimensional lattice, a bounded number of shifts of the tiling is sufficient for every vertex to reside in the center of some annulus.
By combining length functionals over all scales, and replacing the regular tiling by a suitable random family of annuli, we obtain, for every > 0, a reversible random weight : ( ) → ℝ + satisfying (1.22) (intuitively, because of the unit area normalization), and such that almost surely eventually dist , ¯ ( , where := + ˜ .In other words, distances in dist are (asymptotically) increased by power ( − )/2.
Thus (1.23) gives for every > 0, eventually almost surely Taking → 0 yields ¯ .This is carried out formally in Section 4.1.

Annealed vs. quenched subdiffusivity
One can express [ | ( , )] in terms of electrical potentials.Doing so, it is natural to arive at two-sided annealed estimates: where expectation is taken over both the walk and the random network ( , ).Then a standard application of Borel-Cantelli gives that almost surely + (1) , but not an almost sure lower bound.On the other hand, provides that ℳ 1/ + (1) almost surely, which entails − (1) almost surely.In this way, the two exponents and are complementary, allowing one to obtain twosided quenched estimates from two-sided annealed estimates.This is crucial for establishing = 2 / , as the upper bound in (1.19) uses the fully quenched exponent ¯ which, in the setting of Theorem 1.1, arises from the lower bound (1.13) on the (partially) annealed exponent ¯ . We remark on the following strengthening of Theorem 1.1.
Corollary 1.6.Under the assumptions of Theorem 1.1, it additionally holds that = and = .
Proof.We may assume that and exist, and = .From Theorem 1.3 we obtain: The relations ¯ ¯ and ¯ ¯ follow from (1.20), yielding 2 Reversible random weights Throughout this section, ( , ) is a reversible random network satisfying [1/ ] < ∞.

Modulus and effective resistance
For a network and two disjoint subsets , ⊆ ( ), define the modulus where the minimum is over all weights : ( ) → ℝ + , and For ∈ ( ) and 0 < < , define the annular modulus: Note that when is finite, the minimizer in (2.1) exists and is unique (as it is the minimum of a strictly convex function over a compact set).In particular, even when is infinite, this also holds for ( , , ), as we have Denote this minimal weight by ( , , , ) .The standard duality between effective resistance and discrete extremal length [Duf62] gives an alternate characterization of M ( , , ), as follows.
Lemma 2.1.For any finite graph and disjoint subsets , ⊆ ( ), it holds that Hence for any (possibly infinite graph) , all ∈ ( ) and 0 , For a function : ( ) → ℝ, we denote the Dirichlet energy We will make use of the Dirichlet principle (see [LP16, Ch. 2]): When is finite and and when is additionally connected, the minimizer of (2.3) is the unique function harmonic on ( ) \ ( ∪ ) with the given boundary values.
The idea here is that, by (2.5), the balls { ( , ) : ∈ , ( )} tend to cover vertices ∈ ( ) for which vol ( , ) ≈ vol ( , 3 ), as long as is chosen sufficiently large.On the other hand, (2.4) will allow us to bound | ( , 2 ) ∩ , ( )|.Referring to the argument sketched at the end of Section 1.3, we will center an annulus at every ∈ , ( ), and thus we need to control the average covering multiplicity to keep [ ( 0 , 1 ) 2 ] finite.
Since the law of ( ) does not depend on the root, we have the following.
The mass-transport principle (MTP) for a random rooted network ( , , ) asserts that for any nonnegative Borel : Unimodular random networks are precisely those that satisfy the MTP (see [AL07]).
(2.9) Before proving the lemma, let us see that it establishes Theorem 2.4.
For 1, let 2 be the weight guaranteed by Lemma 2.5, and define the random weight , so that (1).
Let us now prove the lemma.
Denote := 5 1+ and, recalling Section 2.1, Then define: : where we have used the fact that ( ) is supported on edges such that ⊆ ( , 2 1+ ).Apply the biased Mass-Transport Principle (2.6) with the functional to conclude from (2.10) that , where we have used = 5 1+ 4 1+ + .Along with independence of the sampling procedure, this yields by definition of ( , ).
Let us use (2.5) with = − 1 to bound where the last line follows from the definition of ( , ).Now choose := 2 , yielding

Markov type and the rate of escape
Our goal now is to prove Theorem 1.5.It is essentially a consequence of the fact that every -point metric space has maximal Markov type 2 with constant (log ) (see Section 3.2 below), and that the random walk on a reversible random graph with almost sure subexponential growth (in the sense of (1.21)) can be approximated, quantitatively, by a limit of random walks restricted to finite subgraphs.

Restricted walks on clusters
Definition 3.1 (Restricted random walk).Consider a network = ( , , ) and a finite subset denote the neighborhood of a vertex ∈ .Define a measure on by where ( ) := { , } ∈ ( ) : { , } ∩ ≠ ∅ is the set of edges incident on .We define the random walk restricted to as the following process { }: For 0, put where we have used the notation ( , ) := { , } ∈ : ∈ .It is straightforward to check that { } is a reversible Markov chain on with stationary measure .If 0 has law , we say that { } is the stationary random walk restricted to .
Proof.Define the transport ( , , , where denotes some Borel measurable subset of G • (recall the definition from Section 2.3).Then the biased mass-transport principle (2.6) gives

Maximal Markov type
A metric space ( , ) has maximal Markov type 2 with constant if it holds that for every finite state space Ω, every map : Ω → , and every stationary, reversible Markov chain { } on Ω, This is a maximal variant of K. Ball's Markov type [Bal92].Note that every Hilbert space has maximal Markov type 2 with constant for some universal (independent of the Hilbert space); see, e.g., [NPSS06, §8].Bourgain's embedding theorem [Bou85] asserts that every -point metric space embeds into a Hilbert space with bilipschitz distortion (log ), yielding the following.
Lemma 3.3.If ( , ) is a finite metric space with = | |, then for every stationary, reversible Markov chain { } on , it holds that
Proof.By assumption, is locally finite, hence ( , Δ) is finite.Thus we may assume that ( ) > 0 for all ∈ ( ) as follows: Define ˆ ( ) = ( ) if ( ) > 0 and ˆ ( ) = 1 otherwise.We may then prove the lemma for ˆ , and observe that because properties (2) and (3) only refer to finite neighborhoods of the root, and ˆ are identical on these neighborhoods, except for a set of zero measure.
Let { : ∈ ( )} be a sequence of independent random variables where is an exponential with rate ( ).Let ∈ [ Δ 4 , Δ 2 ) be independent and chosen uniformly random.For a finite subset ⊆ ( ), write ( ) := ∈ ( ).We need the following elementary lemma.Lemma 3.6.For any finite subset ⊆ ( ), it holds that Proof.A straightforward calculation shows that min{ : ∈ \ { }} is exponential with rate ( \ { }).Moreover, if and are independent exponentials with rates and , respectively, then In other words, we remove edges whose endpoints receive different labels.
With this in hand, we can proceed to our goal of proving Theorem 1.5.

Exponent relations
Let us now prove the nontrivial inequalities in Theorem 1.3.Since we can take > 0 arbitrarily small, this yields ¯ * , completing the proof.

Effective resistance and the Green kernel
Assume again that ( , ) is a reversible random network.
Definition 4.2 (Green kernels).For ⊆ ( ), let := min{ 0 : ∈ }, and define the Green kernel killed off by It By definition, for any > 0, we have that almost surely eventually Therefore almost surely eventually Combined with (4.3), this gives almost surely eventually As this holds for every > 0, it yields the claimed inequality.
Finally, let us prove that the assumptions (1.11) and (1.12) imply ˜ = ˜ 0 in the case > 0. The first part of the argument follows [BCK05, §3.2].The second part uses methods similar to those employed by Telcs [Tel89].
Since the edge conductances only depend on the differences − , the law of the conductances is translation invariant, and thus ( , 0, ) is a reversible random network.Moreover, [1/ 0 ] < ∞ follows from the fact that − is a Gaussian of variance of bounded variance for { , } ∈ ( ).

Lemma 4.16.
There is some = ( ) > 0 such that for every 8 sufficiently large, the following holds.Proof of Theorem 4.15.Fix some > 0, and define := log( ) .Since the events {ℰ : 1 − 1} involve disjoint sets of edges, they are independent, and we have Moreover, the series law for effective resistances gives Thus an application of Borel-Cantelli yields: Almost surely eventually, Since > 0 could be chosen arbitrarily small, we conclude that ˜ = 0.
Here one notes that this conclusion cannot be reached for = 0 because we cannot choose large enough with respect to so as to create a gap between the respective upper and lower bounds in (4.11) and (4.12).Indeed, it is this sort of gap that Telcs defines as "strongly recurrent" (see [Tel01, Def.2.1]), although his quantitative notion (which requires a uniform multiplicative gap with = ( )) is too strong for us, as it entails ˜ > 0.Let us assume that is such that (4.13) holds.Denote by the induced graph on [ ( , 2 )], and consider the sets