Routeing properties in a Gibbsian model

Routeing properties in a Gibbsian model
for highly dense multihop networks

Wolfgang König111TU Berlin, Straße des 17. Juni 136, 10623 Berlin, and WIAS Berlin, Mohrenstraße 39, 10117 Berlin, koenig@wias-berlin.de and András Tóbiás222Berlin Mathematical School, TU Berlin, Straße des 17. Juni 136, 10623 Berlin, tobias@math.tu-berlin.de

WIAS Berlin and TU Berlin, and TU Berlin


(15 February 2019)

Abstract: We investigate a probabilistic model for routeing in a multihop ad-hoc communication network, where each user sends a message to the base station. Messages travel in hops via other users, used as relays. Their trajectories are chosen at random according to a Gibbs distribution, which favours trajectories with low interference, measured in terms of signal-to-interference ratio. This model was introduced in our earlier paper [KT18], where we expressed, in the limit of a high density of users, the typical distribution of the family of trajectories in terms of a law of large numbers.
In the present work, we derive its qualitative properties. We analytically identify the emerging typical scenarios in three extreme regimes. We analyse the typical number of hops and the typical length of a hop, and the deviation of the trajectory from the straight line, (1) in the limit of a large communication area and large distances, and (2) in the limit of a strong interference weight. In both regimes, the typical trajectory approaches a straight line quickly, in regime (1) with equal hop lengths. Interestingly, in regime (1), the typical length of a hop diverges logarithmically in the distance of the transmitter to the base station. We further analyse (3) local and global repulsive effects of a densely populated subarea on the trajectories.


MSC 2010. 60G55; 60K30; 65K10; 82B21; 90B15; 90B18; 91A06

Keywords and phrases. Multihop ad-hoc network, signal-to-interference ratio, Gibbs distribution, message routeing, high-density limit, point processes, variational analysis, expected number of hops, selfish routeing

1. Introduction

In this work, we continue our research [KT18] on a spatial Gibbsian model for random message routeing in a multihop ad-hoc network with device-to-device (D2D) communication. In [KT18] we prepared for an analysis of the qualitative properties of the model by deriving simplifying formulas that describe the situation in a densely populated area in the sense of a law of large numbers. In the present work, we carry out this analysis and describe a number of characteristic properties of the message trajectories. In particular, we are interested in the interplay between probabilistic properties like entropy and energetic properties like interference and congestion and how this interplay influences geometric characteristics like number and lengths of the hops or shapes of the trajectories. Our goal is to identify some rules of thumbs in the relationships between all these quantities in asymptotic regimes in which they become particularly pronounced, like large areas and long trajectories, strong influence of interference, or local regions with a particularly high population. While our previous paper used mainly probabilistic methods, the present paper entirely employs analytic tools.

1.1. The main features of the model

Let us introduce our telecommunication model. The communication area is a bounded set in , and it has a unique base station at the origin . Many users are distributed in according to some measure. Each user sends out a message to the base station along a random multihop trajectory that uses other users as relays and has at most hops. (There is no mobility of the users (nodes) in our model.) We are interested in the joint distribution of all these random message trajectories, conditional on the locations of the users.

Our main idea is to study a trajectory distribution that favours configurations with a high service quality from the viewpoint of interference. Under the distribution, all the message trajectories are stochastically independent. Each individual trajectory is distributed in the following way. A priori, it has a uniform distribution (i.e., it chooses first a hop number and then a -hop trajectory, both uniformly at random), and there is an exponential weight term penalizing interference. This term is the sum of the reciprocals of the signal-to-interference ratios (SIRs) for each hop of the message trajectory. (In Section 6.3.2 we will point out that the penalty term is a certain approximation of the bandwidth used for the multihop transmission.) That is, the total penalty is given to the entire trajectory collection in terms of a probability weight. In the language of statistical mechanics, such a probability measure for collections of trajectories is called a Gibbs distribution. The highest probability is attached to those trajectory families that realize the best compromise between entropy (i.e., probability) and energy (i.e., interference); i.e., the Gibbs distribution respects the transmission properties of the entire system. Note that there is no strict SIR threshold, i.e., hops with bad SIR values are suppressed but not forbidden, similarly to the setting of [FDTT07]. In Section 6.2.1, we comment on a version of the model where hops with low SIR values are excluded.

This model is of snap-shot type without time dependence. Indeed, we assume that all messages are transmitted, relayed, and received at the same time. Further, all users act as transmitters but also as relays and can receive and forward messages, while the base station has only the function of a receiver. According to this, we define SIR in such a way that the interference for any hop of any message is determined by the spatial positions of all users (i.e., the starting points of all message trajectories), analogously to the model considered in [HJKP18]. The independence of message trajectories under our Gibbsian trajectory distribution is a consequence of this choice, and it would not hold in time-dependent versions of the model. We comment on possible ways of introducing a time dimension in the model and their effect on the notion of SIR in Section 6.2.3.

Summarizing, we consider an ad-hoc network with D2D communication in a bounded communication area with a large number of users and a single base station. Nevertheless, we note that we are not aware of a real-world telecommunication network that works according to the routeing policy that we use in this paper. One of our motivations is to explore the physical effect of the penalization of the joint probability of the random paths, which are a priori randomly picked with equal probability: Does the (soft) requirement of a good transmission quality force the trajectories to choose geometrically the shortest route? What hop lengths do they choose? We would like to understand the interplay between entropy and interference-energy and emerging effects.

The idea of an optimal trade-off between entropy and energy is most clearly realized in a certain limiting sense in [KT18, Theorem 1.4], which will be the starting point of the present paper and will be summarized in Section 2. There, we carried out the limit of a high density of users, and we derived a kind of law of large numbers for the “typical” trajectory distribution, i.e., the joint trajectory distribution that has the highest probability under the Gibbs measure. The optimal trajectory collection was obtained as the minimizer of a characteristic variational formula. Roughly speaking, the variational formula is of the form “minimize the sum of entropy and energy among all admissible trajectory families”, see Section 6.3.3.

In fact, in [KT18] we considered an extended version of the above model with another exponential weight term penalizing congestion. This term counts the ordered pairs of incoming hops arriving at the relays in the system. This is certainly an important characteristics of the quality of service, as too high an accumulation of many messages at relays results in a delay. An important property of this term is that it introduces dependence between the trajectories of different messages, unlike the interference term. Hence, this model represents a situation with a centralized choice of all trajectories in the spirit of a common welfare, instead of selfish routeing optimization. In Section 7, we give a game-theoretic discussion of the two weight terms in the exponent in the light of traffic theory; more precisely we ask under what circumstances the optimization of the sum of these two terms can be called selfish or non-selfish. In Section 6.3.4, we also make a connection between this optimization and our model from the viewpoint of stochastic algorithms. According to the results of [M18], realizing our Gibbsian system numerically using Monte Carlo Markov chain methods is on average much more effective than finding the optimum. This gives another motivation for our model.

With the above definition of message trajectories, we only consider uplink communication, i.e., users transmitting messages to the base station. The downlink, i.e., the reversed direction, works very similarly. We believe that all the results of [KT18] as well as the ones of the present paper have an analogue for the downlink with an analogous proof, and we refrain from presenting details.

1.2. Goals

Our goal in the present paper is to understand the global effects that are induced in the Gibbsian system exclusively by entropy and energy into geometric properties of the trajectory collection. As our model depends on various parameters (size and form of the communication area, density of users, choice of the interference term, strength of interference weighting, etc.), this can be done rigorously only in certain limiting regimes, namely:

  1. large communication area and long distances (and large hop numbers),

  2. strong interference penalization, and

  3. high local density of users on a subset of the communication area.

We are interested in geometric properties such as the typical hop lengths, the average number of hops, and the typical shape of the trajectory. In regimes (1) and (2), we expect that the typical trajectories approach straight lines, and in (1) there is an additional question about the typical length of a hop and the number of hops. Here, we would like to understand how the quality of service becomes bad in a large telecommunication area and how many and how large hops the messages would like to take if the constraint on the maximum number of hops is dropped.

However, the regime (3) and our questions here are of a different nature. We would like to determine if the presence of a subarea with a particularly high population density has a significant (positive or negative) impact on the effective use of the relaying system: on the one hand, the trajectories have more available relays in such an area, but on the other hand, the interference achieves high values there. This is a trade-off between entropy and energy that we want to understand.

Let us point out that we are going to work on these questions only in the case where only interference is penalized, but not congestion. We decided this because the description of the minimizer(s) of the variational formula in [KT18, Proposition 1.3] is enormously implicit and cumbersome in general, but reduces, if the congestion term is dropped, to relatively simple formulas that are amenable for analytical investigations [KT18, Proposition 1.5]. In particular, only in this setting we know that the minimizer is unique. We believe that the main qualitative properties persist to the case where also congestion is penalized, as this is purely combinatorial and not spatial. In this paper, the congestion term appears only in modelling discussions. Its formal definition is postponed until Section 6.3.1.

1.3. Our findings

In regimes (1)–(2), we will see that the typical trajectory follows a straight line with exponential decay of probabilities of macroscopic deviations from this shape. Moreover, in regime (1) we will also find simple formulas for the asymptotic number of hops and the average length of a hop, which turns out to be the same for each hop of the trajectory. One of our most striking findings is that, in regime (1), the typical hop length diverges as a power of the logarithm of the distance between the transmitter and the base station, and hence the typical number of hops is sub-linear in the distance. This effect seems to come from the facts that the total mass of the intensity measure of the communication area diverges and that, a priori, i.e., before switching on the interference weight, every message trajectory of a given length has the same weight, even very unreasonable ones that have long spatial detours, e.g., many loops.

However, in regime (3), we encounter different effects. First we see the following global effect on the total number of relaying hops in the entire system: if the communication area is small (in the sense that all the interferences in the system do not vary much), then the total number of relaying hops vanishes exponentially fast in the diverging parameter of the dense population, regardless of the choice of the densely populated subset, as long as it has positive Lebesgue measure. In some cases, we also detect a local effect on the relaying hops if the densely populated subset is very small: we demonstrate that a certain neighbourhood of that subset is definitely unfavourable for relaying hops for practically all the other users. This is a very clear effect coming from the high interference of the densely populated area, which expels the trajectories away.

Some of our results are easy to guess, and the main value of our work is the explicit characterization of the quantities and the derivation of exponential bounds for deviations. We formulated our results in quite simple settings, by putting the communication area equal to a ball and the user density equal to the Lebesgue measure, but it is clear that they can be extended into various directions with respect to more complex shapes and/or user distributions.

Based on our explicit formulas, we also provide simulations in Section 8. They illustrate that most of the effects that we derived analytically in limiting settings, i.e., for large values of the parameters, already appear in a very pronounced way for quite moderate values of the parameters.

1.4. Related literature

The quality of service in highly dense relay-augmented ad-hoc networks has received particular interest in the last years. A multihop network with users distributed according to a Poisson point process with diverging intensity was investigated in [HJKP18]. Using large deviations methods, that paper derives the asymptotic behaviour of rare frustration events such as many users having an unlikely bad quality of service for an unusually long period of time. [HJP18] also describes frustration probabilities in a network, where relays have a bounded capacity, and users become frustrated when their connection to a relay is refused because it is already occupied; see also [HJ17].

One difference between these works and the Gibbsian model of the present paper introduced in [KT18] is that the latter one uses a notion of quality of service for the entire system rather than for single transmissions. In particular, trajectories with bad SIR are a priori not excluded. There is a random mechanism for choosing the message trajectories of all users, given the user locations, and our results hold almost surely with respect to the point process of user locations in the high-density limit. For these results, users need not form a Poisson point process, and they can even be located deterministically [KT18, Section 1.7.4]. This is also a difference from [HJKP18, HJP18, HJ17], where user locations are not fixed and their randomness is (at least partially) responsible for unlikely frustration events.

For literature remarks on the notion and use of SIR, in particular for multiple hops, regarding the choice of a bounded path-loss function, and about the interference penalization term, see Section 6.1 later.

Gibbs sampling was used for various aspects of modelling telecommunication networks, e.g., in [CBK16] for optimal placement of contents in a cellular network, and in [BC12] for power control and associating users to base stations. These Monte Carlo Markov chain methods are used to decrease some kind of cost in the system via a random mechanism, with no easily implementable deterministic methods being available. Our Gibbsian model also has this property if both interference and congestion are penalized. The recent master’s thesis of Morgenstern [M18] investigated the use of a Gibbs sampler or a Metropolis algorithm for an experimental realization of our Gibbsian system; see Section 6.3.4 for a summary.

As for mathematical works about message routeing in interference limited multihop ad-hoc networks, let us mention the papers [BBM11, IV17]. In these works, users are randomly selected as transmitters or receivers in each time slot, the success of transmissions is determined by an SIR constraint, and the main question is about the finiteness of the expected delay and the positivity of the information velocity. Since in the model of the present paper bad SIR values are penalized “softly”, i.e., we do not require that each hop of each message trajectory have a sufficiently large SIR value, further our model does not include a time dimension, our main objects of study are of a different nature.

1.5. Organization of this paper

In Section 2, we present our Gibbsian model and the results of [KT18] that are relevant for the investigations of the current paper.

Each of the following three sections is devoted to one of our three theoretical investigations, which form the core of this paper, i.e., the analysis of the large-distance limit (1) in Section 3, the limit of strong interference penalization (2) in Section 4 and the limit of high local density of users (3) in Section 5. Each of these sections gives the question, the results, the proofs and a discussion in the respective setting.

Section 6 contains modelling discussions and conclusions. Here we discuss the notion of SIR and sketch some possible extensions of the model. Further, we provide motivations for our Gibbsian ansatz in the case when both interference and congestion are penalized.

Section 7 discusses the relevance and properties of our Gibbsian model and the related optimization problem in the light of game-theoretic considerations in traffic theory.

Finally, Section 8 gives numerical plots and studies about qualitative properties of our model.

2. The Gibbsian model and its behaviour in the high-density limit

In this section, we recall the Gibbsian model of [KT18] and its properties in the limit of high density of users. We present the model in Section 2.1, describe its behaviour in the high-density limit in Section 2.2 and comment on the notion of the typical trajectory sent out by a user in Section 2.3. The main objects we will consider in this paper are defined in Sections 2.2 and 2.3, while the nomenclature and interpretation of these objects originate from the preceding Section 2.1.

2.1. The Gibbsian model

We introduce the model that we study in the present paper. This model was introduced in [KT18, Section 1.2.4]; it is a special case of the general model of [KT18]. Here we only consider the case where only interference is penalized and congestion is not. For the definition of the model where also congestion is weighted, we refer the reader to Section 6.3.1.

For any and for any measurable subset of , let denote the set of all finite nonnegative Borel measures on . We write for .

We are working in with fixed. Let be compact, the area of the telecommunication system, containing the origin of . Let be an absolutely continuous measure on with . For , we let be a Poisson point process in with intensity measure . We refer to the as to the users of the wireless network, thus is the number of users in the network.

Now, we introduce message trajectories. Fix and . A message trajectory from to with hops is of the form , where is the transmitter, are the relays and is the receiver. Our modelling assumption is that each user submits exactly one message to along a trajectory (i.e., ). Further, we write for the configuration of all these trajectories. We denote by the set of all such trajectory configurations.

Next, we introduce interference penalization. We choose a path-loss function, which describes the propagation of signal strength over distance. This is a monotone decreasing, continuous function . A typical choice is corresponding to ideal Hertzian propagation, i.e. , for some (see e.g. [GT08, Section II.]). The signal-to-interference ratio (SIR) of a transmission from to in the presence of the users in is defined [HJKP18] as

The sum in the denominator of the right-hand side of (2.1) is the interference. In fact, according to conventional nomenclature, one should say “total received power” instead of interference and “signal-to-total received power ratio” instead of SIR. We discuss this in Section 6.1, where we also comment on the factor in the denominator of (2.1) and on the effect of the boundedness of the path-loss function. In Section 6.2.3 we point out how (2.1) would change if we introduced time dependence in our model.

Let us fix a parameter . Now, for a message trajectory from to , we define

Now, the central object studied in [KT18] is the following Gibbs distribution on the set of configurations of trajectories. For put

This is the Gibbs distribution with a uniform and independent a priori measure (see [KT18, Section 1.2.2] for details), subject to an exponential weight with the sums of the reciprocals of the SIR values of all hops. Here

is the normalizing constant, which is referred to as partition function. Note that is random and defined conditional on , and it is a probability measure on .

2.2. The limiting behaviour of the telecommunication system

We study the above wireless communication system in the high-density limit .

Now we summarize the results of [KT18] that are relevant for the present paper. For , elements of the product space are denoted as . For , the -th marginal of a measure is denoted by , i.e., for any Borel set of .

We assume that the empirical measure of normalized by , i.e., the measure

(2.5)

converges to almost surely in the high-density limit . (We write for the Dirac measure at .) This condition is satisfied e.g. if is increasing; for further details see [KT18, Section 1.7.4]. However, note that is not normalized; its total mass converges towards .

For fixed and for a trajectory collection , we define the empirical measure of all the -hop trajectories of as

(2.6)

This is the main object behind the following analysis; it registers where the main bulk of the trajectories runs. Note that is not normalized. Since each user sends out exactly one message, we have

This assumption can be relaxed, see Section 6.2.2 for a discussion about this. Note that for and , we have

We denote by a random variable with distribution . Since as and (2.2) holds for any , subsequential limits of in the coordinatewise weak topology are easily seen to be of the form with , satisfying

cf. [KT18, Section 3.4]. For such , we define the following analogue of (2.2) with replaced by its limit :

The key result [KT18, Proposition 1.5, parts (3), (4)] about the limiting behaviour of the telecommunication system that we will use this paper is the following.

Proposition 2.1 (Law of large numbers for the empirical measures).

Let and . Then, almost surely with respect to , as , the distribution of converges coordinatewise weakly to the collection of measures , where

and the normalizing function is defined as

so that (2.2) holds.

Let us note that the case is trivial because in this case, all messages are transmitted directly to the base station, and thus converges coordinatewise weakly to , where .

In the limiting measure (2.1), the starting points of the -hop message trajectories, , are chosen according to the measure and the th relays according to the measure for all , exponentially weighted by the limiting interference penalization term .

In Section 6.3.3 we will explain that the measures (2.1) form the unique minimizer of a characteristic variational formula; this property will not be used for deriving the main results of the present paper. For further details about the limiting behaviour of the system, see [KT18, Sections 1.6, 1.7].

2.3. Interpretation of the limiting trajectory distribution

The purpose of the present paper is to make further qualitative assertions about the “typical” trajectory from a given transmission site to the origin, after having taken the limit . A definition of the “typical” trajectory as a random variable is not immediate, due to the nature of this setting. In the present paper, we will focus on the probability measure on given by its Radon–Nikodym derivative

(2.12)

with respect to . This function is the main object of our study in the present paper. We normalized in such a way that . According to Proposition 2.1,

(2.13)

where we recall (2.2). We will use the convention that the 0th coordinate of is the one corresponding to and the th is the one corresponding to , for . This way, the marginal is a measure on .

We note that also the measure carries interesting information about the system. Indeed, in [KT18, Section 1.3] it was explained that, at a position , the typical number of incoming hops of a user at is Poisson distributed with parameter , and the total mass is the amount of relaying hops in the entire system, with the convention that it is zero if every message hops directly into without any relaying hop. Part of our analysis will also be devoted explicitly to , see Section 5.

3. Large communication areas with large transmitter–receiver distances

This section is devoted to the analysis of the highly dense telecommunication system described in Section 2.2 in regime (1), i.e., in the limit of a large communication area coupled with a large distance of the user from the base station. In Section 3.1, we present our main results and in Section 3.2 we prove them. Section 3.3 includes discussions related to this regime.

3.1. The typical number, length, and direction of hops in a large-distance limit

In this section, the main object of interest is the typical shape of the trajectory from a certain site to the origin, in particular the typical length of any of the hops, the number of hops, and the spatial progress of the trajectory, in particular whether or not it runs along the straight line or how strongly it deviates from it. We will answer these questions for the special choice that is a closed ball around the origin, is the Lebesgue measure on , and the path-loss function corresponds to ideal Hertzian propagation so that , that is, for some .

Furthermore, in order to obtain a transparent picture and to derive a neat result, we will have to assume that the starting site of our trajectory is far away from the origin. In such a setting, it is plausible to expect that, as the radius of the ball tends to infinity, a proportion of users that tends to one takes the same order of magnitude of number of hops. This also gives information about the typical length and direction of each hop in large but still compact communication areas.

We will see that this setting exhibits the interesting property that the typical number of hops diverges to infinity as the distance of the user from tends to infinity, however, in a sublinear way, more precisely, like the distance divided by a power of its logarithm. Second, using the asymptotics of the value of this largest summand in (2.1), one can conclude about the typical length of the hops and about how much they deviate from the straight line between the transmitter and the receiver . In our specific setting, we will be able to obtain precise and explicit asymptotics for all these quantities.

We denote the radius of the communication area by , and we recall that is the maximal hop number. We consider the limit of large and large . We consider one user placed at with a distance from the origin being large, such that , but . (We write “” if the quotient of the two sides stays bounded and bounded away from zero.) Then one can say that for large , is a “typical” location of a user in , chosen uniformly at random.

In our first result, Theorem 3.1, we examine the “typical” number of hops of a trajectory from to as a random variable under the marginal distribution on . According to (2.13), in the present setting, this is given by

(3.1)

where is the volume of the unit ball in , and we recall that . It is a priori not clear what the relation between and in the limiting setting should be in order to obtain interesting assertions. Nevertheless, while depends on via the identity , we observe that the terms can also be defined for analogously to (3.1) for . Thus, it will be our first task to find the asymptotics of without any reference to . We encounter a large deviation principle on a quite surprising scale.

Theorem 3.1 (Large deviations for the hop number).

Fix . Then, in the limit with , for any choice of ,

if , (3.2)
if  (3.3)
if . (3.4)

where we recall that .

The upper bound in (3.3) follows from the convexity of and a comparison between the functionals and .

Note that Theorem 3.1 identifies the growth of on the scale for on the scale ; indeed, the second and third line rule out small and large values of on that scale, and the first line identifies the precise dependence on the prefactor. In more technical terms, satisfies, with , a large deviation principle on the scale with rate function . It is easily seen that this rate function has a unique minimizer

(3.5)

with minimum value

As a consequence, we have the following law of large numbers.

Corollary 3.2.

In the limit with , any maximizer of satisfies

Further, if for at least one such maximizer for all sufficiently large , then we have

If is smaller than all the minimizers, then the asymptotics of depend on those of rather than on , and (3.2) has to be adapted accordingly. We note that (3.2) requires only a lower bound on , and in Corollary 3.2, could be equal to for each ; see Section 3.3.3 for a discussion about allowing arbitrary many hops in our model. (3.2) says that the asymptotic logarithmic behaviour of on scale coincides with the one of the single maximal summand . Formulated in terms of the marginal distribution of on the length of the path from to , since the behaviour of the Lebesgue measure restricted to is subexponential in in the large-distance limit that we are considering, we have that

tends to zero exponentially fast on the scale for all . In Section 3.3.1 we give an explanation of how these scales arise.

In the proof of the lower bound of (3.2), considering a uniform hop length distribution was sufficient, i.e., hops along the same straight line directed from to with length each. We now show, again in terms of a large deviations estimate on the scale , that macroscopic deviations from this optimal hop length on the scale have extremely small probability. For a finite set , we write for the cardinality of .

Proposition 3.3.

For and , let

(3.7)

Then, in the limit with , for ,

(3.8)

In words, the probability that there are hops , for some index set , in the trajectory of relays such that their average hop length deviates from the optimal hop length on that scale, decays exponentially fast to zero on the scale . (In the denominator of the summands in (3.7), we have removed the factor in order to simplify notation.)

We presented the results of this section for the path-loss functions of the form , , which makes the notation in the proofs less heavy. However, these assertions require only two properties of : the integrability of over and the convexity of , see Section 3.3.2.

The proofs of Theorem 3.1, Corollary 3.2 and Proposition 3.3 are carried out in Sections 3.2.1, 3.2.2 and 3.2.3, respectively. A discussion about these results and their proofs can be found in Section 3.3.1.

Certainly, our results of this section hold for much more general communication areas , not only for balls. Essential for our approach is only that a – in every space dimension diverging – neighbourhood of the straight line between and is contained in in the limit considered. The parameter appearing in the rate function goes back to our assumption that the volume of grows like the -th power of ; however, other powers than in are also possible by putting other geometric assumptions on .

3.2. Proof of the results of Section 3.1

All the three results of Section 3.1 tell about the limit with , where has Euclidean norm . Throughout this section, we will use the notation for this limit and refer to it as “our limit”.

3.2.1. Proof of Theorem 3.1

Our strategy for proving the three assertions (3.2), (3.3) and (3.4) is the following. First we verify the lower bound in (3.2). Then we prove (3.3) and afterwards (3.4), and we combine these two proofs in order to conclude the upper bound in (3.2).

Proof of (3.2), lower bound. Let us first consider satisfying just . We obtain a lower bound for defined in (3.1) by restricting the -integral to the ball with radius one around for . Then, eventually, for . Note that , where we recall that . Hence, for any , eventually,

where in the first step we used that tends to infinity in our limit. This gives

where the second inequality holds eventually, since . Now an elementary optimization on shows that is the relevant scale. Then, in the particular case that for some , carrying out the limit and making afterwards, we have

which is the lower bound in (3.2).

Proof of (3.3). This proof uses that is convex and that the numerator can be well approximated by for sufficiently many . These arguments lead to the following lemma.

Lemma 3.4.

Let . If for all sufficiently large, then eventually in our limit,

holds simultaneously for all with and .

Proof.

Let us now define an auxiliary function such that and in our limit. Fix . The idea is to pick sufficiently large so that

Let us assume that we are given a trajectory with , and . Let us define the index of the last hop outside :

which we want to understand as if there is no such hop. Let be sufficiently large so that and (3.2.1) holds. Then we have

(3.11)
(3.12)
(3.13)
(3.14)

In (3.11) we used the fact that lie in and therefore (3.2.1) can be applied for the numerator of each with . Next, (3.12) is an application of Jensen’s inequality for , and (3.13) uses the following fact. Either , in which case

or , and thus

In both cases, the argument in is , and we can write the term in terms of the -norm and the first step in (3.14) also follows. Hence, we have derived (3.4). ∎

By Lemma 3.4, for any , we have eventually in our limit, under the assumptions of the lemma

Now, let and such that (in particular eventually). Then,

(3.17)

for all , and thus

which is (3.3).

Proof of (3.4). Note that for any , we have