Table Of ContentCritical dynamical heterogeneities close to continuous second-order glass transitions
Saroj Kumar Nandi1, Giulio Biroli1, Jean-Philippe Bouchaud2, Kunimasa Miyazaki3, David R. Reichman4
1 Institut de Physique Th´eorique, CEA/DSM/IPhT-CNRS/URA 2306 CEA-Saclay, F-91191 Gif-sur-Yvette, France
2 Capital Fund Management, 23 rue de l’Universit´e, 75007 Paris, France.
3 Institute of Physics, University of Tsukuba, Tsukuba 305-8571, Japan and
4 Department of Chemistry, Columbia University, 3000 Broadway, New York, NY 10027, USA.
(Dated: January 28, 2014)
We analyse, using Inhomogenous Mode-Coupling Theory, the critical scaling behaviour of the
dynamical susceptibility at a distance ǫ from continuous second-order glass transitions. We find
4
that the dynamical correlation length ξ behaves generically as ǫ−1/3 and that the upper critical
1
dimension is equal to six. More surprisingly, we find activated dynamic scaling, where ξ grows
0
with time asln2t exactly at criticality. All theseresults suggest adeep analogy between theglassy
2
behaviour of attractive colloids or randomly pinned supercooled liquids and that of the Random
n Field Ising Model.
a
J
7 Several recent studies have revealed that the proper- the “Inhomogeneous” MCT (IMCT) formalism, which
2
ties ofthe glasstransitioncanbe drasticallymodified by was developed by some of us [8] to describe dynamical
suitably tuning some control parameters. In the case of heterogeneities at the usual MCT transition. We shall
]
n thecolloidalglasstransition,anattractiveinteractionon obtainthemean-fieldvaluesofthecriticalexponents,the
n
topofthe hard-sphererepulsioncanchangethe dynami- uppercriticaldimensionandderivethecriticalbehavior,
-
s calbehaviour andleadto glass-glasstransitions and log- which turns out to be very different from the usual one.
di arithmic relaxation [1]. This behaviour is also expected We find in particular activated dynamic scaling, which
. for glassy liquids in porous media [2]. For generic glass- strongly bolsters the relationship with the RFIM [3, 5].
t
a forming liquids, it was recently predicted [3, 4] that ran- InordertograspthemainpropertiesoftheA3critical
m
domly pinning a fraction of particles would transmute point, it is useful to focus on the mean-field Landau-
- the glass transition in a continuous second-order phase like potential V(f;ε), called the Franz-Parisi (FP) po-
d
n transition akin to that of the Random Field Ising model tential in the present context [9]. The arguments ε and
o (RFIM) [3–5]. There are strong theoretical indications f are, respectively, the vector of all control parameters
c that these phenomena are in fact all related to the ex- that can be tuned (e.g. the temperature and the frac-
[ istence of a new kind of glassy critical point, first found tion of pinned particles) and the glassy (non-ergodic)
2 within Mode-Coupling Theory (MCT) as a higher-order order parameter, which measures how far the dynam-
v singularity [6]. The physical contexts in which it ap- ics can displace the system away from its initial config-
3 pears are quite different: for attractive colloids it is a uration. For usual glass transitions, the FP potential
5
terminal point of a glass-glass transition line, for glass- has a unique minimum f = 0 at high temperatures; it
2 0
3 forming liquids either pinned or trapped in porous me- corresponds to a complete loss of memory of the initial
. diaitcorrespondstothelocuswheretheMode-Coupling condition as normal in a liquid. A secondary minimum
1
transition and the Ideal Glass transition lines merge. In appears for f = f∗ > 0 below a certain transition tem-
0
4 theformercasetheglasstransitionlinestopsatthisnew perature Tc, see Fig. 1. An important achievement of
1 critical point [3], whereas in the latter it carries on and the last decades was to establish that T actually co-
c
v: becomes continuous [2]. Remarkably, in all these physi- incides with the MCT transition, where locally stable,
i calsituations,activatedprocessesnotdescribedbyMCT long-lived amorphous structures, corresponding to the
X areeithersuppressed,sincetheregionwheretheyappear secondary minimum of V(f;ε), appear. Roughly speak-
r shrinks to zero for glass-forming liquids, or not relevant ing, the connection between the FP potential and MCT
a
since the system is in the glass phase for attractive col- maybe expressedasV′(f;ε)= f [f],where [f]
loids. Hence,MCTmightbecomequantitativelyaccurate 1−f −Fε Fε
is the memory function of the MCT equations (The full-
inthesesituations. Thedynamicalbehaviourofthe two-
fledged MCT calculation deals with wave-vector depen-
point functions at this new glassy critical point, that we
dent order parameters f , we will return to this below)
q~
will call A3 using G¨otze’s terminology, was predicted by
[10]. An A3 critical point corresponds to the merging
MCTcomputations[6]andconfirmedlaterbothnumeri- of the minima f and f∗ [11, 12]. It is of co-dimension
0
cally and experimentally in colloids [1]. The static prop-
2, much as the liquid-gas critical point, i.e. one needs
ertiesofthe fluctuations ofthe overlapfieldbetweentwo
to tune at least two control parameters to reach it, as
equilibrium configurations were recently investigated in
found for systems with quenched pinning sites (see inset
[3,5]and,usingsimulations,in[7]. Acompletetheoryof
of Fig. 1) and for hard sphere systems, where one tunes
dynamical correlations is however still lacking. The aim
the short-range attractive interactions and the density.
of this paper is to develop such a theory by extending
Technically, the existence of an underlying thermody-
2
T ǫ|| critical point) occurs when the secondary minimum of
T >Tc (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) V(f) just appears, implying Vc(2) = 0 (see Fig. 1). The
ǫ) ǫ⊥ next order singularity (A3) occurs when V(3) concomi-
, c
f
( tantly vanishes as well (see Fig. 1). Looking for the new
V c
location f∗ of the minimum away from the transition,
T <Tc one finds, to leading order in ǫ= ~ε: f∗ fc √ǫ, as is
A2 critical point familiar for the A2 case and f∗ |f| √3−ǫ in ∼the generic
c
− ∼
A3 critical point A3 case. There is however a subtlety here: since the A3
f
pointrequiresatleasttwoparameterstobevariedsimul-
taneously,oneneedstoincludethecasewherethechosen
FIG. 1: Schematic presentation of the FP potential. The
trajectory in parameter space is precisely perpendicular
second minimum that develops below the transition point
to ~ V′, in which case one finds again the weaker sin-
becomes flat as the usual MCT (A2) transition point is ap- ∇ε c
proachedfrombelow. ThetwominimacoalesceoneattheA3 gularity f∗ −fc ∼ √ǫk, where ǫk is the distance to the
point. Inset: schematic phase diagram obtained for random critical point along that special direction. The motiva-
pinningglasstransitions. Inthetemperature-pinningfraction tionforthenotationǫ stemsfromtheliquid-glassphase
k
planetheMCTtransitionlineendsinanA3criticalpoint[3]. diagram, sketched in the inset of Fig. 1 in the case of
The two special directions mentioned in the text are shown.
random pinning glass transitions [3]. In general A3 is
the terminal point of the line of A2 critical points. The
special direction found above is the one tangent to the
namicalformulationhas beenextremely useful to under-
A2 line. In consequence, we also introduce the notation
stand that the MCT transition is necessarily accompa-
ǫ for the magnitude of the component of ~ε perpendicu-
nied by the divergence of a length scale, which governs ⊥
lar to the A2 line (i.e. parallel to ~ V′).
the spatial extent over which dynamical fluctuations are ∇ε c
ThemainideaofIMCT[8]istoperturbthesystemwith
correlated, a feature that was hard to anticipate within
asmallspatiallyperiodicexternalpotential cos(~q ~x),
the original framework of G¨otze et al. This diverging 0
∝ ·
whose spatial profile varies over the length-scale 1/q .
length scale is in fact a direct consequence of the van- 0
ishing of the curvature V′′(f∗;ε) of the FP potential at The characteristic value of q0∗ at which the external per-
turbationstartstoactdifferentlyfromauniform,q =0,
T (see Fig. 1)[13–15]. The IMCT formalismallowedus 0
c
perturbation allows one to obtain the correlation length
to make a series of precise predictions about the space-
of dynamical heterogeneities as ξ = 1/q∗. Since all the
time scaling of dynamical heterogeneities in supercooled 0
physicsofthe slowingdownis governedbythe vanishing
liquids close to T [8]. One finds in particular that the
c
dynamical correlation length diverges as T T −1/4 as of the curvature of the FP potential, the crucial point is
c
| − | toworkouthowtheperiodicperturbation(ofzeromean)
thecriticalpointisapproached. AlthoughtheIMCTpre-
changes this curvature. Because the system is rotation-
dictions are only expected to be correctfar enough from
ally invariant, it is reasonable to assume that the extra
T below 8 dimensions, many general predictions appear
c contributiontothecurvatureis q2. Therefore,onehas:
to be confirmed,sometimesquantitatively,by largescale ∼ 0
computersimulations,see[16]andrefstherein. Thesede-
1
velopments, and others, strongly support a quantitative V′′(f∗,ε) V(3)(f∗ f )+ V(4)(f∗ f )2+~ V(2) ~ε+Γq2.
≈ c − c 2 c − c ∇ε c · 0
theory of supercooled liquids built using the mean-field
scenarioas a starting point, muchas Curie-Weiss theory
ClosetoanA2criticalpoint, V(3) =0andf∗ f √ǫ,
provides a foundation for the modern theory of critical c 6 − c ∼
which shows that the characteristic value of q beyond
phenomena [17, 18]. 0
whichtherelaxationtimesubstantiallychangesis ǫ1/4,
In the following we first explain our results in an in- ∼
leading to ξ ǫ−1/4, in agreement with the result of [8].
formalandsimple waybasedonthe behaviourofthe FP ∼
Upon approaching an A3 critical point, V(3) = 0, lead-
potential;wethensketchthecompleteIMCTderivation. c
WewilldenoteasV(n) then-thderivativeofV(f;ε)with ing to ξ ∼ ǫ⊥−1/3 in the generic case, and to ξ ∼ ǫk−1/2
respect to f (V′ = V(1), etc.), and ~εV the gradient in the special case where ǫ⊥ =0. These results are fully
of V with respect to the parameters.∇The expansion of confirmed,andmademoreprecise,bytheIMCTanalysis
V′(f;ε) around the transition point f =f , ~ε=0 reads, that we now briefly present.
c
with δf =f f : The IMCT formalism starts from an exact equa-
c
− tion for the inhomogeneous dynamic structure factor
1 1
V′(f;ε) ≈ Vc(2)δf + 2Vc(3)δf2+ 3Vc(4)δf3 hFo(mq1o,gqe2n,eto)u=s ehxρtqe1rn(ta)lρfi−eql2d(0u)(iqin):the presence of an in-
0
+ ~ V′ ~ε+ ~ V(2) ~εδf +...,
∇ε c · ∇ε c ·
∂F(q1,q2,t) +Ω F(q ,q ,t)+ tM(q ,k,t t′)
where we have used that by definition, at the transition, ∂t q1 1 2 k 0 1 −
V′(fc;0) ≡ 0. The standard MCT transition (the A2 ×∂F(k∂,tq′2,t′)dt′ =TuP(q1,Rq2,t), (1)
3
14 0.3 eigenvalues of Cq;k(q0) can be computed using pertur-
bation theory. Due to rotational invariance, one finds
122× q00.2 that all eigenvalues of Cq;k(0) are shifted by an amount
χ q00.1 q2. The smallest eigenvalue of I C(q ) is therefore
10 ∝ 0 − 0
equaltoαǫ 2/3+Γq2,whereαandΓarenumbers,and
0 | ⊥| 0
(t)]q=008 0 q0 (l5o0g t)2 100 tǫh⊥is6=ve0r.yTsmhaellsoeliugteinovnaloufe,Eaqn.d(t2h)uwsirlelabbdes:dominated by
χ
log[10 46 εεε⊥⊥⊥===666××× 111000−−−345 χq0(k,t→∞)≈ α|ǫh⊥ψ|L2|/S3i+ψkRΓq02, (3)
ε⊥=6× 10−6 where ψL,R are the left and right largest eigenvectors
2 ε⊥=6× 10−7 of Cq;k at criticality. From this expression, one directly
ε⊥=6× 10−8 demonstrates the existence of a diverging susceptibility
0 −40 −20 0 20 40 60 80 100 120 140 and a diverging length scale within MCT, which is inti-
log t
10 mately due to the vanishing of the curvature of the FP
potential. ClosetoanA3point,thislengthscalediverges
FIG. 2: Numerical results obtained by solving the schematic as ǫ −1/3, as announced above. The time dependent
⊥
F13 equations. χq0=0 as a function of time approaching the ana|lys|is is more cumbersome and will be presented in
A3criticalpoint,whichisindeedfoundtorescaleas|ǫ |−2/3
⊥ detail elsewhere [23]. The final result is:
times a function of |ǫ |1/6 × lnt. Inset: Scaling collapse,
⊥
fcorritidcaifflietrye,natsqp0readnicdtetd, forfomq02χthcqe0 tahseoaryf.unction of q0ln2t at χq0(k,t) ≃ 1+V⊥Γ((kq)ξξ⊥2 )2Gk √lnξt ,q0ξ⊥ , ǫ⊥ 6=0,
0 ⊥ (cid:18) ⊥ (cid:19)
where Ωq1 ≡q12kBT/Sq1 is a frequency term, M(q1,k,t) χq0(k,t) ≃ 1+VkΓ((kq)0ξξk2k)2G⊥ lnξtk,q0ξk!, ǫ⊥ =0,
is the memory kernel and (q ,q ,t) contains all
u 1 2
T
the terms generated due to the external potential. where V (k) are certain functipons, and ξ = ǫ −1/3,
k,⊥ ⊥ ⊥
The dynamical susceptibility is defined as χq0(q1,t) = ξk = ǫk −1/2, as indeed anticipated by the simp|le|argu-
δF(q ,q +q ,t)/δu(q ) . This object obeys a lin- | |
1 1 0 0 u→0 ments above.
|
earequation(thatwedonotwritehere,see[8])obtained
The scaling functions (u,v) are such that in the
k,⊥
bytakingthederivativeofEq. (1)withrespecttou(q ). G
0 limit ǫ 0 and q = 0 (corresponding to a uni-
k,⊥ 0
In the long-time limit, this linear equation reads: →
form perturbation), the dynamical susceptibility is well
defined. This imposes (u,v = 0) u4, and there-
k,⊥
G ∼
δk,q−Cq;k(q0) χq0(k,t→∞)=S(q,q0), (2) fore, at criticality, χc0(k,t) ∼ ln4t, a result that can be
k (cid:18) (cid:19) obtained directly by taking the derivative of the scal-
X
ing form of F(k,t) close to an A3 critical point, see
whereS(q,q )isanonsingularsourcetermandC (q )
0 q;k 0 [11], and that is compatible with the simulation results
is a q dependent matrix that can be fully computed in
0 of attractive colloids reported in [19]. Away from criti-
terms of the memory kernel of the model, see [8]. Note cality, the ln4t behaviour only persists up to a time τ
ξ
that χq0(k,t → ∞) is nothing else than the variation of such that lnτξ ξk (or lnτξ √ξ⊥ along the special
the non-ergodic parameter f =F(k,t ) due to the ∼ ∼
k line ǫ = 0.). Note that the exponential dependence of
→∞ k
external potential. In order to analyse Eq. (2) we recall p
the relaxation time as a function of the distance from
(see [11]) that C (0)=(1 f )2∂ [ f ]/∂f . The
q;k k q,ε k k the critical point was already known from the critical
− F { }
propertiesoftheoperatorδ C (0),whichisakinto
k,q q;k behaviour of the dynamical structure factor, see [11].
−
V′(f;ε),arereportedin[11]: atdistanceǫfromthetran-
Another interesting limit is when the system is critical
sition,onefindsf =fc+(1 fc)2g ,whereg √3ǫψR
k k − k k k ∝ k ǫk,⊥ = 0 and perturbed at a non-zero spatial frequency,
and ψkR is the (right) zero mode of I−C(0) evaluated at q0 6= 0. In order to retain a non-trivial dynamics, one
the transition point. This scaling holds when approach- must now have (u 0,v ) = g(u2v), where
k,⊥
G → → ∞
ing the A3 critical point in any directiobn other than the g(x) is a certain function behaving as x2 for small x,
one parallel to the line of usual A2 transitions that ap- and saturating to a constant for x . This leads
proach the A3 point. In this case one finds gk ∝ √ǫψkR. to: χcq0(k,t) = q0−2g(q0ln2t), which →show∞s that at crit-
Away from criticality, the smallest eigenvalue of the ma- icality, the dynamical length grows without bound, as
trix I C(0) is not exactly zero: the deviations are of ξ (t) ln2t [20]. This should be compared with the
order,−respectively, ǫ2/3 and ǫ close to an A3 point, de- ccorres∼ponding result for the A2 point, where ξ ta/2,
⊥ k c ∼
pendingobnthedirectionofapproachtocriticality. Com- where a is the MCT exponent for the β-regime. As the
ing back to our original problem, we remark that the A3 point is approached,the value of a tends to zero and
4
the power law crosses over to a logarithmic behaviour. fluctuationsisshifteddowntoavalued =6. Thisresult
c
Note that there is no α-regime at the A3 point, contrar- isindeedqualitativelyconsistentwiththe factthatsome
ily to the usual phenomenology of the A2 transition. In aspectsofdynamicalheterogeneity(suchasbimodalityof
the latter case, the q dependence of χ for q ξ−1 particledisplacementdistributions)aresuppressedasone
crossesoverfromq−20inthe shorttime,qβ0-regim0e,≫toq−4 tunes,viathe introductionofshort-rangedattractions,a
0 0
inthe longtime, α-regime. Forthe A3transition,onthe supercooled hard-sphere suspension to a regime domi-
other hand, only the q−2 behaviour survives. We have nated by higher-order singularities [27]. More numerical
0
checked all these results numerically by solving exactly work on this and other aspects of our theory would be
thedynamicalequationobeyedbyχ (t)intheso-called welcome, using simulations of attractive hard-spheres in
q0
schematic limit where all k dependence is discarded. We higher dimensions, along the lines of [28].
have chosen the F13 model for which the memory kernel Finally, the alert reader will have recognized, both
is ε[f] = ǫ1f +ǫ3f3. The salient features of the above from the evolution of the FP potential shown in Fig. 1
F
scaling predictions for χq0(t) are confirmed in Fig. 2. and the value of the exponent β = 1/3 in generic direc-
Froma purely phenomenologicalpoint of view,the most tions and β = 1/2 in a special direction, that the A3
importantpointsare: (1)thegrowthofχq0 isalogarith- critical point is akin to an Ising transition, where ǫk is a
mic function of the relaxation time thus implying that magnetic-field-likeperturbationandǫ isatemperature-
⊥
dynamical heterogeneities increase much slower close to likeperturbation. Thebehaviourofthe correlationfunc-
an A3 critical point than close to an A2 one; this might tion G (r) r4−d and the corresponding value d = 6
4 c
∼
explain the numerical data of [21]; (2) the shape of χq0 point towards the universality class of the RFIM, in line
is markedly different from the one found at an ordinary withpreviousstatictreatments[3,5]. Indeed,theA2line
MCT transition. In particular, both the maximum and isanalogoustothe spinodallineoftheRFIM[24],which
the long-timelimitofχq0 divergeasthe transitionisap- terminates at the A3 RFIM critical point. Remarkably,
proached, at variance with what happens close to an A2 thelogarithmicbehaviourofthecorrelationfunctionand
point,wherethelong-timelimitofχq0 remainsbounded. ofthe dynamicalcorrelationlengththatwefoundisusu-
All the above results should be only be valid in high allyamanifestationofactivated events,whichareindeed
enoughdimensions. Inordertoassestheeffectofcritical expected for the RFIM at criticality, but were thought
fluctuationsonmean-fieldresultsonehastofocusonthe tobeimpossibletograspwithinanMCTformalism[30].
4-point density correlations often called G (r), which is On the other hand, as mentioned above, cooperative ac-
4
related(in Fourier space)to the square of the dynamical tivated processes not described by MCT are either sup-
susceptibilityχ definedabove(see[22]forafulljustifi- pressedornotrelevantatanA3criticalpoint,thusopen-
q0
cationofthis relation). Since χ behavesasq−2 atcrit- ingthepossibilitythataboved>6MCTindeedcaptures
q0 0
icality, G (r) isfoundtodecayas1/rd−4 uptodistances the correct behaviour. Curiously, however, the dynam-
4
oforderξ. Thisallowsonetoestimatetheintensivefluc- ical behaviour of the RFIM in high dimensions has not
tuations of the order parameterf∗ in a regionof size ξd, beenworkedoutyet. Usualscalingargumentsleadingto
which is found to be δf2 ξ−(4−d)/2 = ǫν(4−d)/2, logτ ξθ (whereθisthestiffnessexponent)areexpected
h i ∼ ∝
which mustbe comparedto f∗ f ǫβ. Since β =ν = to break down above d = d 5.1 [29]. Understand-
p − c ∼ DR ≃
1/3 for A3 transitions (or β =ν =1/2 along the special ingtowhatextentdynamicscalingholdsaboved and
DR
direction) we conclude that the upper critical dimension comparing the actual RFIM dynamical behaviour above
d below which criticalfluctuations change the nature of sixdimensiontothepredictionofMCTnearanA3point
c
the transition is d = 6, instead of d = 8 as found for is certainly a topic worth a further studies.
c c
usual MCT transitions [22, 24]. Therefore, in physical GB and SN acknowledgesupport from the ERC grant
dimensions d = 3, one expects that these critical fluctu- NPRGGLASS. DRR acknowledges NSF CHE-1213247.
ations will considerably affect the above predictions, at KM acknowledges KAKENHI No. 24340098,25103005,
leastcloseenoughtothecriticalpoint. Therelativeinflu- and the JSPS Core-to-CoreProgram. We wish to thank
ence of these fluctuations, and the quantitive size of the C. Cammarota, V. Krakoviack, M. Sperl and G. Tarjus
Ginzburg region where critical fluctuations are strong, for helpful discussions.
are expected to depend on the model. But one conse-
quence of these fluctuations that is physically relevant is
the violationof the Stokes-Einsteinrelation,relating the
relaxation time of the system to the diffusion constant
of probe particles. Simulations of thermal and ather- [1] F. Sciortino and P. Tartaglia, Adv. in Phys. 54 471
(2005).
mal[25,28]systems appeartoconformtothe prediction
[2] V.Krakoviack,Phys.Rev.Lett.94,065703(2005);Phys.
d =8 for usual MCT transitions, yielding, for example,
c Rev. E 75, 031503 (2007); Phys. Rev. E 84, 050501(R)
a Stokes-Einstein violation exponent that vanishes lin-
(2011).
earlyas8 d[26]. Ouraboveanalysisdemonstratesthat [3] C. Cammarota, G. Biroli, PNAS 109, 8850 (2012); J.
−
near an A3 singularity the upper critical dimension of Chem. Phys.138, 12A547 (2013).
5
[4] C. Cammarota, Europhysics Letters 101 56001 (2013). [20] Notethatξ (τ )≡ξ ,infullagreementwiththeabove
c ξ k,⊥
[5] S.Franz, G. Parisi, JSTAT(2013) P11012. relation between τ and ξ.
ξ
[6] K.A.Dawson,G.Foffi,M.Fuchs,W.G¨otze,F.Sciortino, [21] R. L. Jack, C. J. Fullerton, Phys. Rev. E 88, 042304
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[10] The connection between the derivative of theFP poten- Phys. J. E. (2011) 34 102.
tialandthelargetimevalueoftheMCTmemory kernel [25] J. D. Eaves and D. R. Reichman, PNAS 106, 15171
isonlyqualitative.Infact f −F [f]andV′(f;ε)have (2009).
1−f ε
thesameshape,whichiswhatweneedforourdiscussion, [26] G.BiroliandJ.-P.Bouchaud,J.Phys.: Condens.Matter
butthey are not equal. 19 205101 (2007).
[11] W. G¨otze, Complex dynamics of glass-forming liquids, [27] D. R. Reichman, E. Rabani and P. L. Geissler, Journal
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[14] G. Biroli and J.-P. Bouchaud, Europhysiscs Letters 67, [30] It is intriguing that MCT predicts near the A3 point a
21 (2004). logarithmic behaviour for both the correlation function
[15] S. Franz, A. Montanari, Journal of Physics A, 40 F251 and for the dynamical correlation length, ξ ∼ln2t. This
(2007). corresponds to the so-called Sinai diffusion law, which
[16] K.Kim,S.Saito,K.Miyazaki,G.Biroli,D.R.Reichman, governsthedynamicsofdomainwallsintheRFIMalong
J. Phys. Chem. B 117, 13259 (2013). the branches of a tree. As noted above, the logarithmic
[17] A. Andreanov, G. Biroli and J.-P. Bouchaud, Euro- behaviour turns into a power law behaviour ξ ∼ ta/2
physicsLett. 88 16001 (2009). in the presence of a field, with a continuously varying
[18] S. Franz, H. Jacquin, G. Parisi, P. Urbani, F. Zamponi, exponent, exactly as for the Sinai model. Whether this
PNAS109, 18725 (2012). analogy is accidental or reveals a deep and interesting
[19] P. Charbonneau, D. R. Reichman, Phys. Rev. Lett. 99, physical feature of the MCT formalism remains to be
135701 (2007) seen.
