text: − − p ϕm(u)s = ϕm n(p )ϕm(u)ϕm n(s ) the word p is a proper preﬁx of m − m n n m ϕm n(p ) or vice versa. Moreover, p is not empty since it contradicts again n m − the point (iii). Suppose p is a non-empty proper preﬁx of ϕm n(p ). It implies m − − n there exists a word z such that p z = ϕm n(p ). If ϕm n(p ) is a non-empty m n −n proper preﬁx of p , then we may ﬁnd a word z such that ϕm n(p )z = p (see m n m Figure 4). a u b pm ϕm(u) sm − − = ϕm(v) ϕm n(p ) ϕm(u) ϕm n(s ) n n pn ϕn(u) sn = ϕn(v) v Figure 3: The ﬁrst arrangement from the proof of Lemma 7. Therefore, in both cases, the word ϕm(u) is a preﬁx of z(cid:96) for some integer (cid:96). Since | | | | | | ϕm(u) ϕm(u) ϕm(u) | | > {| | | − |} > | | > (cid:15)(k), z max p , ϕm n(p ) ϕm(a) m n (cid:98) (cid:99) we deduce that z (cid:15)(k) is factor of ϕm(u). ∞ As lim (cid:15)(k) = + , the D0L-system is repetitive. → ∞ k + Lemma 8. In any PD0L system there is a constant C such that all factors over bounded letters longer than C have a synchronizing point. Proof. The statement is trivial for non-pushy D0L systems, hence we consider ∈ A a pushy one. Clearly, there exist an integer n such that for all c we have | | | | ≥ 0 ϕm(c) = ϕm+1(c) for every m n. Let u be a factor over bounded letters (cid:107) (cid:107) · | | only of length at least L = 3 ϕn+1 w where w is the axiom of the D0L 0 0 system. This implies that u appears in the sequence E(G) = (w ) ≥ in w for i i 0 k k > n + 1. Let (p, w, s) be an interpretation of u. Since u is a factor of w such that | | ∈ A∗ ∈ Ak k > n + 1 and w > L, there must be words x, y and v + such that k w = xϕn(v)y and | | − | | (cid:107) (cid:107) | | − | | (cid:107) (cid:107) ϕ(x) p < ϕn+1 and ϕ(y) s < ϕn+1 . 5 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: p

text: ϕm(u)s

text: ϕm

text: n(p

text: )ϕm(u)ϕm

text: n(s

text: the

text: word

text: p

text: is

text: a

text: proper

text: preﬁx

text: of

text: m

text: m

text: n

text: n

text: m

text: ϕm

text: n(p

text: or

text: vice

text: versa.

text: Moreover,

text: p

text: is

text: not

text: empty

text: since

text: it

text: contradicts

text: again

text: n

text: m

text: the

text: point

text: (iii).

text: Suppose

text: p

text: is

text: a

text: non-empty

text: proper

text: preﬁx

text: of

text: ϕm

text: n(p

text: It

text: implies

text: m

text: n

text: there

text: exists

text: a

text: word

text: z

text: such

text: that

text: p

text: z

text: ϕm

text: n(p

text: If

text: ϕm

text: n(p

text: is

text: a

text: non-empty

text: m

text: n

text: −n

text: proper

text: preﬁx

text: of

text: p

text: then

text: we

text: may

text: ﬁnd

text: a

text: word

text: z

text: such

text: that

text: ϕm

text: n(p

text: )z

text: p

text: (see

text: m

text: n

text: m

text: Figure

text: a

text: u

text: b

text: pm

text: ϕm(u)

text: sm

text: ϕm(v)

text: ϕm

text: n(p

text: ϕm(u)

text: ϕm

text: n(s

text: n

text: n

text: pn

text: ϕn(u)

text: sn

text: ϕn(v)

text: v

text: Figure

text: The

text: ﬁrst

text: arrangement

text: from

text: the

text: proof

text: of

text: Lemma

text: Therefore,

text: in

text: both

text: cases,

text: the

text: word

text: ϕm(u)

text: is

text: a

text: preﬁx

text: of

text: for

text: some

text: integer

text: Since

text: ϕm(u)

text: ϕm(u)

text: ϕm(u)

text: z

text: max

text: p

text: ϕm

text: n(p

text: ϕm(a)

text: m

text: n

text: we

text: deduce

text: that

text: z

text: is

text: factor

text: of

text: ϕm(u).

text: As

text: lim

text: the

text: is

text: repetitive.

text: k

text: Lemma

text: In

text: any

text: system

text: there

text: is

text: a

text: constant

text: C

text: such

text: that

text: all

text: factors

text: over

text: bounded

text: letters

text: longer

text: than

text: C

text: have

text: a

text: synchronizing

text: point.

text: Proof.

text: The

text: statement

text: is

text: trivial

text: for

text: non-pushy

text: systems,

text: hence

text: we

text: consider

text: A

text: a

text: pushy

text: one.

text: Clearly,

text: there

text: exist

text: an

text: integer

text: n

text: such

text: that

text: for

text: all

text: c

text: we

text: have

text: ϕm(c)

text: for

text: every

text: m

text: n.

text: Let

text: u

text: be

text: a

text: factor

text: over

text: bounded

text: letters

text: only

text: of

text: length

text: at

text: least

text: L

text: w

text: where

text: w

text: is

text: the

text: axiom

text: of

text: the

text: system.

text: This

text: implies

text: that

text: u

text: appears

text: in

text: the

text: sequence

text: E(G)

text: (w

text: in

text: w

text: for

text: i

text: i

text: k

text: k

text: n

text: Let

text: (p,

text: w,

text: s)

text: be

text: an

text: interpretation

text: of

text: u.

text: Since

text: u

text: is

text: a

text: factor

text: of

text: w

text: such

text: that

text: A∗

text: Ak

text: k

text: n

text: and

text: w

text: L,

text: there

text: must

text: be

text: words

text: x,

text: y

text: and

text: v

text: such

text: that

text: k

text: w

text: xϕn(v)y

text: and

text: ϕ(x)

text: p

text: and

text: ϕ(y)

text: s

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Ground Truth gorithm could check for paths to a person’s known home Nodes along ﬁnal paths address at the end of predicted paths. Second, at the be- ginnings of paths drivers move from local roads to larger arteries, so the algorithm could also attempt to simply take 2 Miles the shortest paths to major roads if there are too many am- 3 km biguous turn choices at the beginning of a path. When accepted for publication, we will open source our work, which contains tools to get the needed map data from Open Street Maps, and implementations of our algorithms. We will also include a subset of the collected traces that has been explicitly authorized to be made publicly available on e CRAWDAD or similar archival sites. d u atit 8. CONCLUSIONS L We have demonstrated an attack against user location pri- vacy that uses speed data from driving traces and an initial starting location. This problem seems simple in concept, but is diﬃcult in execution largely because turn detection with speed data is often ambiguous. For instance, if someone breaks and then accelerates they could have turned or they could have slowed down because the person in front of them was turning. We have presented the design and implemen- tation of a novel elastic pathing algorithm and used a large Longitude corpus of real-world traces to show its accuracy. Our trials showed that the algorithm is fast, and could also be used for Figure 6: An example path where the algorithm cor- real-time tracking of people just based on their speed data. rectly follows most of a path but predicts the wrong Our results show that with a large corpus of data an entity turn when the speed trace runs out of features. can errode individuals’ privacy and gain information about typical routes and destinations. In 20% of individual traces only require the mileage information from vehicles [28, 31, we predicted the correct trip end point to within 500 meters. 26]. PAYD coverage provided by Hollard Insurance Com- This means that a location visited daily can be identiﬁed in pany collects mileage information and driver style informa- about a week; locations visited on a weekly basis could be tion [29]. Companies like PayGo Systems collect driving identiﬁed with slightly more than a month of data. zone information instead of ﬁne-grained location, besides ba- Variations in daily routines are easier to identify than end- sic information including mileage and minutes of use [32]. points. Even when the algorithm does not correctly ﬁnd the The best practice for such monitoring systems is to only proper end point it does correctly follow much of the actual record relevant data. For example, devices could check speed travelled path. If that predicted path breaks from a normal data but only send a notiﬁcation when an even of interest routine, such as going home after work, then a malicious was detected. In general, this approach could be applied to entity may be able to either deduce where the person could other data collection methods, including GPS. have gone with some foreknowledge of the person’s habits 7.2 Limitations and Improvements or take advantage of their altered routine. In time, we expect that improvements will be made to our Our main limitation is the diﬃculty distinguishing be- initial approach. The data, once collected, does not go away tween similar roads. When a vehicle stops at an intersec- and a continually improving path prediction algorithm may tion and then accelerates slowly a turn or straight path are plumb more private information from these traces. These both possible, so our algorithm relies upon later features in traces should be treated as any other private information the speed trace to determine which of the possible routes instead of being handled as already anonymized data. Our was correct. At the ends of paths there are simply too few work is not directed against any company or organization features to make this distinction. At the beginning of paths – experiences have shown that even well-intentioned uses of this may simply cause there to be too many possible paths to data can result in losses of privacy so we wish to highlight sort through. These problems are exacerbated in residential potential dangers of this type of data collection. areas, where repeated, similar intersections are common. Usage-based insurance has been introduced in at least 39 We notice that travel through residential areas tends to states of the United States, and more are approving the ap- occur at either ends of paths rather than during the mid- proach because it has been claimed to be privacy-preserving. dle, because traveling through a residential area is unlikely Our work shows that speed-based data collection erode user to be the fastest route to a destination. Fixing mistakes privacy of policy holders over time. We do not claim that in residential areas by incorporating additional knowledge insurance companies are violating policy holder privacy in about the area may greatly improve our algorithm’s success this way. However, the general principle of any privacy pre- since we had little diﬃculty reconstructing paths along ma- serving data collection is to collect only the data that is jor roads, but tend to make mistakes once the path leaves necessary for a particular application, and no more. major roads. An example of this is in Figure 6. There are two ways these errors could be addressed. First, the street addresses of customers are available so the al- ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Ground

label: paragraph

font: default

text: Truth

text: gorithm

text: could

text: check

text: for

text: paths

text: to

text: a

text: person’s

text: known

text: home

text: Nodes

text: along

text: ﬁnal

text: paths

text: address

text: at

text: the

text: end

text: of

text: predicted

text: paths.

text: Second,

text: at

text: the

text: be-

text: ginnings

text: of

text: paths

text: drivers

text: move

text: from

text: local

text: roads

text: to

text: larger

text: arteries,

text: so

text: the

text: algorithm

text: could

text: also

text: attempt

text: to

text: simply

text: take

text: Miles

text: the

text: shortest

text: paths

text: to

text: major

text: roads

text: if

text: there

text: are

text: too

text: many

text: am-

text: km

text: biguous

text: turn

text: choices

text: at

text: the

text: beginning

text: of

text: a

text: path.

text: When

text: accepted

text: for

text: publication,

text: we

text: will

text: open

text: source

text: our

text: work,

text: which

text: contains

text: tools

text: to

text: get

text: the

text: needed

text: map

text: data

text: from

text: Open

text: Street

text: Maps,

text: and

text: implementations

text: of

text: our

text: algorithms.

text: We

text: will

text: also

text: include

text: a

text: subset

text: of

text: the

text: collected

text: traces

text: that

text: has

text: been

text: explicitly

text: authorized

text: to

text: be

text: made

text: publicly

text: available

text: on

text: e

text: CRAWDAD

text: or

text: similar

text: archival

text: sites.

text: d

text: u

text: atit

text: CONCLUSIONS

text: L

text: We

text: have

text: demonstrated

text: an

text: attack

text: against

text: user

text: location

text: pri-

text: vacy

text: that

text: uses

text: speed

text: data

text: from

text: driving

text: traces

text: and

text: an

text: initial

text: starting

text: location.

text: This

text: problem

text: seems

text: simple

text: in

text: concept,

text: but

text: is

text: diﬃcult

text: in

text: execution

text: largely

text: because

text: turn

text: detection

text: with

text: speed

text: data

text: is

text: often

text: ambiguous.

text: For

text: instance,

text: if

text: someone

text: breaks

text: and

text: then

text: accelerates

text: they

text: could

text: have

text: turned

text: or

text: they

text: could

text: have

text: slowed

text: down

text: because

text: the

text: person

text: in

text: front

text: of

text: them

text: was

text: turning.

text: We

text: have

text: presented

text: the

text: design

text: and

text: implemen-

text: tation

text: of

text: a

text: novel

text: elastic

text: pathing

text: algorithm

text: and

text: used

text: a

text: large

text: Longitude

text: corpus

text: of

text: real-world

text: traces

text: to

text: show

text: its

text: accuracy.

text: Our

text: trials

text: showed

text: that

text: the

text: algorithm

text: is

text: fast,

text: and

text: could

text: also

text: be

text: used

text: for

text: Figure

text: An

text: example

text: path

text: where

text: the

text: algorithm

text: cor-

text: real-time

text: tracking

text: of

text: people

text: just

text: based

text: on

text: their

text: speed

text: data.

text: rectly

text: follows

text: most

text: of

text: a

text: path

text: but

text: predicts

text: the

text: wrong

text: Our

text: results

text: show

text: that

text: with

text: a

text: large

text: corpus

text: of

text: data

text: an

text: entity

text: turn

text: when

text: the

text: speed

text: trace

text: runs

text: out

text: of

text: features.

text: can

text: errode

text: individuals’

text: privacy

text: and

text: gain

text: information

text: about

text: typical

text: routes

text: and

text: destinations.

text: In

text: of

text: individual

text: traces

text: only

text: require

text: the

text: mileage

text: information

text: from

text: vehicles

text: we

text: predicted

text: the

text: correct

text: trip

text: end

text: point

text: to

text: within

text: meters.

text: PAYD

text: coverage

text: provided

text: by

text: Hollard

text: Insurance

text: Com-

text: This

text: means

text: that

text: a

text: location

text: visited

text: daily

text: can

text: be

text: identiﬁed

text: in

text: pany

text: collects

text: mileage

text: information

text: and

text: driver

text: style

text: informa-

text: about

text: a

text: week;

text: locations

text: visited

text: on

text: a

text: weekly

text: basis

text: could

text: be

text: tion

text: Companies

text: like

text: PayGo

text: Systems

text: collect

text: driving

text: identiﬁed

text: with

text: slightly

text: more

text: than

text: a

text: month

text: of

text: data.

text: zone

text: information

text: instead

text: of

text: ﬁne-grained

text: location,

text: besides

text: ba-

text: Variations

text: in

text: daily

text: routines

text: are

text: easier

text: to

text: identify

text: than

text: end-

text: sic

text: information

text: including

text: mileage

text: and

text: minutes

text: of

text: use

text: points.

text: Even

text: when

text: the

text: algorithm

text: does

text: not

text: correctly

text: ﬁnd

text: the

text: The

text: best

text: practice

text: for

text: such

text: monitoring

text: systems

text: is

text: to

text: only

text: proper

text: end

text: point

text: it

text: does

text: correctly

text: follow

text: much

text: of

text: the

text: actual

text: record

text: relevant

text: data.

text: For

text: example,

text: devices

text: could

text: check

text: speed

text: travelled

text: path.

text: If

text: that

text: predicted

text: path

text: breaks

text: from

text: a

text: normal

text: data

text: but

text: only

text: send

text: a

text: notiﬁcation

text: when

text: an

text: even

text: of

text: interest

text: routine,

text: such

text: as

text: going

text: home

text: after

text: work,

text: then

text: a

text: malicious

text: was

text: detected.

text: In

text: general,

text: this

text: approach

text: could

text: be

text: applied

text: to

text: entity

text: may

text: be

text: able

text: to

text: either

text: deduce

text: where

text: the

text: person

text: could

text: other

text: data

text: collection

text: methods,

text: including

text: GPS.

text: have

text: gone

text: with

text: some

text: foreknowledge

text: of

text: the

text: person’s

text: habits

text: Limitations

text: and

text: Improvements

text: or

text: take

text: advantage

text: of

text: their

text: altered

text: routine.

text: In

text: time,

text: we

text: expect

text: that

text: improvements

text: will

text: be

text: made

text: to

text: our

text: Our

text: main

text: limitation

text: is

text: the

text: diﬃculty

text: distinguishing

text: be-

text: initial

text: approach.

text: The

text: data,

text: once

text: collected,

text: does

text: not

text: go

text: away

text: tween

text: similar

text: roads.

text: When

text: a

text: vehicle

text: stops

text: at

text: an

text: intersec-

text: and

text: a

text: continually

text: improving

text: path

text: prediction

text: algorithm

text: may

text: tion

text: and

text: then

text: accelerates

text: slowly

text: a

text: turn

text: or

text: straight

text: path

text: are

text: plumb

text: more

text: private

text: information

text: from

text: these

text: traces.

text: These

text: both

text: possible,

text: so

text: our

text: algorithm

text: relies

text: upon

text: later

text: features

text: in

text: traces

text: should

text: be

text: treated

text: as

text: any

text: other

text: private

text: information

text: the

text: speed

text: trace

text: to

text: determine

text: which

text: of

text: the

text: possible

text: routes

text: instead

text: of

text: being

text: handled

text: as

text: already

text: anonymized

text: data.

text: Our

text: was

text: correct.

text: At

text: the

text: ends

text: of

text: paths

text: there

text: are

text: simply

text: too

text: few

text: work

text: is

text: not

text: directed

text: against

text: any

text: company

text: or

text: organization

text: features

text: to

text: make

text: this

text: distinction.

text: At

text: the

text: beginning

text: of

text: paths

text: experiences

text: have

text: shown

text: that

text: even

text: well-intentioned

text: uses

text: of

text: this

text: may

text: simply

text: cause

text: there

text: to

text: be

text: too

text: many

text: possible

text: paths

text: to

text: data

text: can

text: result

text: in

text: losses

text: of

text: privacy

text: so

text: we

text: wish

text: to

text: highlight

text: sort

text: through.

text: These

text: problems

text: are

text: exacerbated

text: in

text: residential

text: potential

text: dangers

text: of

text: this

text: type

text: of

text: data

text: collection.

text: areas,

text: where

text: repeated,

text: similar

text: intersections

text: are

text: common.

text: Usage-based

text: insurance

text: has

text: been

text: introduced

text: in

text: at

text: least

text: We

text: notice

text: that

text: travel

text: through

text: residential

text: areas

text: tends

text: to

text: states

text: of

text: the

text: United

text: States,

text: and

text: more

text: are

text: approving

text: the

text: ap-

text: occur

text: at

text: either

text: ends

text: of

text: paths

text: rather

text: than

text: during

text: the

text: mid-

text: proach

text: because

text: it

text: has

text: been

text: claimed

text: to

text: be

text: privacy-preserving.

text: dle,

text: because

text: traveling

text: through

text: a

text: residential

text: area

text: is

text: unlikely

text: Our

text: work

text: shows

text: that

text: speed-based

text: data

text: collection

text: erode

text: user

text: to

text: be

text: the

text: fastest

text: route

text: to

text: a

text: destination.

text: Fixing

text: mistakes

text: privacy

text: of

text: policy

text: holders

text: over

text: time.

text: We

text: do

text: not

text: claim

text: that

text: in

text: residential

text: areas

text: by

text: incorporating

text: additional

text: knowledge

text: insurance

text: companies

text: are

text: violating

text: policy

text: holder

text: privacy

text: in

text: about

text: the

text: area

text: may

text: greatly

text: improve

text: our

text: algorithm’s

text: success

text: this

text: way.

text: However,

text: the

text: general

text: principle

text: of

text: any

text: privacy

text: pre-

text: since

text: we

text: had

text: little

text: diﬃculty

text: reconstructing

text: paths

text: along

text: ma-

text: serving

text: data

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text: 1.2 b=0.20 data b=0.30 data b=0.40 data 1 0.8 R 0.6 0.4 0.2 0 3 4 5 6 7 8 9 10 N FIG. 15. Quenched b = 0.20, 0.30, 0.40 ratio versus N. The ratio either approaches a constant value signiﬁcantly diﬀerent from zero, or shows an increasing trend with N. Clearly there is a ≥ dramatic change for b 0.40. gives c = 0.0438(22), c− = 1.909(57), χ2/d.o.f. = 4.88 (5.15) 0 2 We attempted other more general forms of N dependence; none of these reduced the χ2/d.o.f.. However, there is a clear decrease with N to a small constant in the large N limit. Thus the eigenvalue distribution becomes to a very good approximation uniform in the large N limit. It will be argued in Section below that this is an indication of unbroken center symmetry. This is also consistent with old results such as [2] that place b = 0.10 below the transition. By contrast, for b = 0.20, 0.30, 0.40 we cannot ﬁt any smooth function to the data, so it is not possible to extrapolate to large N. The most that can be said is that the ratio tends ∼ to a small value for b = 0.20 and a relatively large value 0.2 for b = 0.30. For b = 0.40 the ∼ ratio is monotonically increasing at large N, reaching quite large values 1.0. For purposes of comparison, Ref. [2] examined N = 5 and found ﬂuctuations in the free energy between ≥ quadratic and quartic behavior for what is b 0.15 in our language. They interpreted this as evidence for symmetry breaking. This suggests that the irregular behavior we see in 20 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: data

text: data

text: data

text: R

text: N

text: FIG.

text: Quenched

text: b

text: ratio

text: versus

text: N.

text: The

text: ratio

text: either

text: approaches

text: a

text: constant

text: value

text: signiﬁcantly

text: diﬀerent

text: from

text: zero,

text: or

text: shows

text: an

text: increasing

text: trend

text: with

text: N.

text: Clearly

text: there

text: is

text: a

text: dramatic

text: change

text: for

text: b

text: gives

text: c

text: c−

text: We

text: attempted

text: other

text: more

text: general

text: forms

text: of

text: N

text: dependence;

text: of

text: these

text: reduced

text: the

text: However,

text: there

text: is

text: a

text: clear

text: decrease

text: with

text: N

text: to

text: a

text: small

text: constant

text: in

text: the

text: large

text: N

text: limit.

text: Thus

text: the

text: eigenvalue

text: distribution

text: becomes

text: to

text: a

text: very

text: good

text: approximation

text: uniform

text: in

text: the

text: large

text: N

text: limit.

text: It

text: will

text: be

text: argued

text: in

text: Section

text: below

text: that

text: this

text: is

text: an

text: indication

text: of

text: unbroken

text: center

text: symmetry.

text: This

text: is

text: also

text: consistent

text: with

text: old

text: results

text: such

text: as

text: that

text: place

text: b

text: below

text: the

text: transition.

text: By

text: contrast,

text: for

text: b

text: we

text: cannot

text: ﬁt

text: any

text: smooth

text: function

text: to

text: the

text: data,

text: so

text: it

text: is

text: not

text: possible

text: to

text: extrapolate

text: to

text: large

text: N.

text: The

text: most

text: that

text: can

text: be

text: said

text: is

text: that

text: the

text: ratio

text: tends

text: to

text: a

text: small

text: value

text: for

text: b

text: and

text: a

text: relatively

text: large

text: value

text: for

text: b

text: For

text: b

text: the

text: ratio

text: is

text: monotonically

text: increasing

text: at

text: large

text: N,

text: reaching

text: quite

text: large

text: values

text: For

text: purposes

text: of

text: comparison,

text: Ref.

text: examined

text: N

text: and

text: found

text: ﬂuctuations

text: in

text: the

text: free

text: energy

text: between

text: quadratic

text: and

text: quartic

text: behavior

text: for

text: what

text: is

text: b

text: in

text: our

text: language.

text: They

text: interpreted

text: this

text: as

text: evidence

text: for

text: symmetry

text: breaking.

text: This

text: suggests

text: that

text: the

text: irregular

text: behavior

text: we

text: see

text: in

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