Extreme-Value Distributions and the Freezing Transition of Structural Glasses

Michele Castellana1, ⇤

1Joseph Henry Laboratories of Physics and Lewis–Sigler Institute for Integrative Genomics,Princeton University, Princeton, New Jersey 08544

We consider two mean-field models of structural glasses, the random energy model (REM) andthe p-spin model (PSM), and we show that the finite-size fluctuations of the freezing temperatureare described by extreme-value statistics (EVS) distributions, establishing an unprecedented con-nection between EVS and the freezing transition of structural glasses. For the REM, the freezing-temperature fluctuations are described by the Gumbel EVS distribution, while for the PSM thefreezing temperature fluctuates according to the Tracy-Widom (TW) EVS distribution, which hasbeen recently discovered within the theory of random matrices. For the PSM, we provide an ana-lytical argument showing that the emergence of the TW distribution can be understood in terms ofthe statistics of glassy metastable states.

PACS numbers: 02.50.-r , 02.10.Yn, 64.70.Q-

The problem of characterizing the maximum of a setof random variables, also known as extreme-value statis-tics (EVS), has attracted considerable interest for sev-eral decades now. EVS has found a remarkable numberof applications in several scientific fields, including engi-neering [1], finance [2], and biology [3]. Moreover, EVShas raised particular interest in physics [4], describingthe phenomenology of several systems of general inter-est, such as dynamical [5], and disordered [6] systems.In particular, EVS has been recently shown to play arole in the critical regime of spin glasses, i.e. disor-dered uniaxial magnetic materials like Fe

0.5Mn0.5TiO

3

and Eu0.5Ba

0.5MnO3

[7, 8]: In this Letter, we establish anovel connection between EVS and a completely differentclass of solid materials. These systems, known as struc-tural glasses, are liquids–like o-Terphenyl and Glycerol–that have been cooled fast enough to avoid cristallisation[9].

Since its very first development, the theory of struc-tural glasses has continuously drawn the attentionof the scientific community: Understanding the low-temperature behavior of these systems and the natureof their glassy phase is still one of the deepest unsolvedquestions in condensed-matter theory [10]. In particular,the existence of a freezing transition in structural glasseshas been the subject of an ongoing debate for the lastfew decades [11]. The development of exactly solvablemodels [12, 13] mimicking the phenomenology of struc-tural glasses showed that such a transition does exist ona mean-field level and, later on, further studies suggested[11, 14] that a freezing transition might occur also forrealistic, non-mean-field [15] systems.

We consider two well-established [11, 16] mean-fieldmodels of the freezing transition of structural glasses, therandom energy model (REM) [12] and the p-spin model(PSM) [13], and we study the disorder-induced fluctua-tions of the critical temperature arising when the systemsize is large but finite. We show that for the REM thefluctuations of the critical temperature are described by aEVS distribution of independent variables: The Gumbeldistribution [1]. For the PSM, the finite-size fluctuationsof the critical point are described by a EVS distribution ofcorrelated variables, the Tracy-Widom (TW) distribution

[17]. The TW distribution has been recently discovered inthe theory of random matrices, and it describes randomfluctuations in a variety of physical systems [18, 19].

Random energy model - Let us start by considering thesimplest model of a structural glass exhibiting a freezingtransition: The random energy model (REM) [12]. Here,the REM will serve as an illustrative model to show therole played by EVS in the structural-glass freezing tran-sition. The REM is defined as a system of N Ising spinsSi = ±1: An energy E[

~S] is assigned to every spin con-figuration ~S, and the energies E ⌘ {E[

~S]} are indepen-dent and identically distributed (IID) Gaussian randomvariables with zero mean and variance N/2. In the ther-modynamic limit, the REM has a freezing phase transi-tion: There is a critical value of the energy ec, which isis the lowest value of e such that the number of states~S with energy E[

~S] = Ne is exponentially large in thesystem size. The inverse critical temperature �c is deter-mined from the threshold energy ec by the temperature-energy relation �c = �2ec, which can be obtained bycomputing the Legendre transform of the partition func-tion [12]. Let us now introduce a critical temperature fora REM with a finite number of spins and let us studyits sample-to-sample fluctuations. If the system size Nis sufficiently large, the threshold energy ec E of an en-ergy sample E coincides with the lowest energy value,i.e. Nec E = min~S E[

~S]. It follows [20] that for large Nthe threshold energy is ec E = ec � �

2Nplog 2

, where � isa random variable distributed according to the Gumbeldistribution: P(� x) = exp(� exp(�x)). It is easyto show that the Gumbel distribution describes not onlythe statistics of the ground state [20], but also the fluc-tuations of the critical temperature. A natural way tointroduce a finite-size critical temperature is to extendthe temperature-energy relation �c = �2ec to systemswith finite sizes [7, 8, 21]: We set �c E = �2ec E , where�c E is the finite-size critical temperature of sample E .Putting this definition together with the above expres-sion for ec E , we obtain the expression for the finite-sizecritical temperature �c E = �c +

�N

plog 2

. This shows thatthe finite-size fluctuations of the critical temperature ofthe REM are described by a EVS distribution, the Gum-

2

�16

�12

�8

�4

0

0.5 1 �c 128 1.5 2

�c

B

�

N = 16

N = 32

N = 64

N = 128

N = 256

FIG. 1: Binder cumulant B of the p-spin model with p = 3as a function of the inverse temperature � for system sizesN = 16, 32, 64, 128, 256 (in red, blue, brown, black, greenrespectively), critical temperature �cN for N = 128, andinfinite-size critical temperature �c.

bel distribution, and that the width of the critical region�c E � �c scales with the system size as 1/N . Given thatthe REM is a highly simplified model for a structuralglass [11, 15], a natural question to ask is whether EVScould play a role in other structural-glass models morerealistic than the REM. We will address this question inthe following Section.p-spin Model - In this Section we will consider the

mean-field p-spin model (PSM) [12, 13]. The PSM cap-tures some fundamental physical features of structuralglasses that are absent in the REM [11]: For example,in the PSM energy minima are no longer single configu-rations ~S like in the REM, but they also allow for smallvibrations around the minimum [11]. Moreover, the PSMreproduces the onset of a dynamical slowing down at lowtemperatures typical of structural glasses [16]. In whatfollows, we will focus on the PSM with p = 3 [22], whichis given by N Ising spins Si = ±1 with HamiltonianH[

~S] = �p3

N

Pi<j<k JijkSiSjSk, where Jijk are IID ran-

dom variables equal to ±1 with equal probability. Unlikethe REM, the problem of studying sample-to-sample fluc-tuations of the critical temperature of the PSM cannot beaddressed analytically. Hence, we have studied the PSMnumerically by means of Monte Carlo simulations com-bined with the parallel-tempering algorithm [23]. Eventhough the CPU time for the PSM increases with the sys-tem size as N3 [22], we exploited the binary form of thecouplings to use an efficient asynchronous multispin cod-ing method [24], allowing us to study extensive systemsizes and numbers of disorder samples. Specifically, westudied systems with N = 16, 32, 64, 128, 256 spins and anumber of disorder samples 2.6⇥ 10

2 S 1.3⇥ 10

5.In the thermodynamic limit, the PSM is known [11]

to have a mixed first/second order phase transition:Given two independent replicas ~S1, ~S2, this phase transi-

0.01

0.1

1 1.2 �c 128 �cJ 1.6 1.8 2

hQ2

i�hQ

i2

�

0.01

0.1

1 1.2 �c 128 �cJ 1.6 1.8 2

hQ2

i�hQ

i2

�

hQ2iJ � hQiJ2

hQ2iJ � hQi2J

FIG. 2: Average order-parameter fluctuations hQ2iJ �hQiJ

2(in red), and single-sample order-parameter fluctua-

tions hQ2iJ � hQi2J (in blue) as functions of the inverse tem-perature � for the p-spin model with p = 3 and N = 128. Thevalue of �c 128 is taken from Fig. 1, and the inverse criticaltemperature �cJ is determined by Eq. (1) (in brown).

tion can be described in terms of their mutual overlapQ ⌘ 1/N

PNi=1

S1

i S2

i . In the high-temperature phase� < �c = 1.535 [22], the two replicas explore expo-nentially many energy minima. Since these minima arerandom states not related by any symmetry, differentminima have zero mutual overlap [11, 16], and one hashQ2i = hQi = 0, where hi is the Boltzmann average, and

denotes the average over disorder samples {Jijk} ⌘ J .In the low-temperature phase � > �c, the two repli-cas are both trapped in a few low-lying energy min-ima, and thus they develop a nonzero mutual overlap:hQi > 0, hQ2i � hQi

2

> 0. Let us now introduce a finite-size critical temperature for the PSM. In the first place,the above discussion of the phase transition of the PSMshows that the infinite-volume critical point �c is thevalue of the temperature at which the order-parameterfluctuations (OPF) arise: hQ2i � hQi

2

= 0 if � . �c,hQ2i � hQi

2

> 0 if � & �c. To introduce a critical tem-perature �cJ of a finite-size PSM with couplings J , werecall that the Binder cumulant B ⌘ 1/2(3�hQ4i/hQ2i

2

)

of a finite-size PSM has a minimum at a given tem-perature: This temperature, which we will denote by�cN , is the temperature at which critical OPF arise [22],and it is shown in Fig. 1. It follows that the quantityhQ2i(�cN ) � hQi(�cN )

2

represents the average OPF atthe critical point. Given these average critical OPF, itis natural to identify the finite-size critical temperature�cJ of sample J as the value of � for which the OPF ofthe sample hQ2iJ (�)� hQiJ (�)2 are equal to the aboveaverage critical value

hQ2iJ (�cJ )� hQiJ (�cJ )

2

= hQ2i(�cN )� hQi(�cN )

2

.(1)

The definition (1) of finite-size critical temperature isdepicted in Fig. 2. As shown in the top inset of Fig. 3, thevariance �2

�N ⌘ �2

cJ � �cJ2 of the critical-temperature

distribution is a decreasing function of the system size N ,

3

10

�4

10

�3

10

�2

10

�1

�4 �2 0 2 4

p(xJ)

xJ

10

�4

10

�3

10

�2

10

�1

�4 �2 0 2 4

p(xJ)

xJ

0

0.2

0.4

�2 0 2

p(xJ)

xJ

0

0.2

0.4

�2 0 2

p(xJ)

xJ

0.05

0.1

0.2

16 32 64 128 256

��N

N

0.05

0.1

0.2

16 32 64 128 256

��N

N

Tracy-Widom

Gumbel

Gaussian

N = 16

N = 32

N = 64

N = 128

�� N

f(N)

FIG. 3: Distribution p(xJ ) of the normalized critical temper-ature xJ for the p-spin model with p = 3 and system sizesN = 16, 32, 64, 128 (in red, blue, brown, black respectively),and Tracy-Widom, Gumbel and Gaussian distributions, allwith zero mean and unit variance (in black). The plot hasno adjustable parameters, and it is in logarithmic scale toemphasize the shape of the distributions on the tails. Topinset: Width �� N of the distribution of the finite-size criticaltemperature as a function of the system size N (in red), andfitting function f(N) = aN�� (in black), with fitting expo-nent � = 0.45± 0.04. Bottom inset: Same plot as in the mainpanel in linear scale.

in particular ��N ⇠ N��, with � = 0.45± 0.04. Indeed,��N represents the width of the critical region of a systemwith size N : As the system size gets larger and larger, thewidth ��N shrinks, and the finite-size critical tempera-ture converges to the infinite-size value �c (see Supple-mental Material) [7, 8, 21, 25]. Given that that the dis-tribution of �cJ obeys the above scaling with the systemsize N , one can expect [8, 21] that the distribution of thenormalized critical temperature xJ ⌘ (�cJ ��cJ )/��N ,converges to a finite limiting shape as the system sizegoes to infinity. This claim is supported by the numer-ical data shown in the main panel and bottom inset ofFig. 3, showing that the distribution of xJ appears toconverge to a limiting distribution for large N . In con-trast to the REM, the shape of this limiting distributiondoes not seem to be the Gumbel distribution, but anotherEVS distribution recently discovered in the theory of ran-dom matrices, the Tracy-Widom (TW) distribution [17],as shown in the main panel and bottom inset of Fig. 3 andas suggested by statistical-hypothesis tests (see Supple-mental Material). Interestingly, the TW distribution hasbeen recently shown to play a role in describing the criti-cal behavior of disordered systems: For example, the TWdistribution characterizes the average number of minimain a simple model of a random-energy landscape closeto its freezing transition [25], and the finite-size fluctua-tions of the critical temperature in mean-field spin glasses[7, 8].

We will now provide an analytical argument to giveinsight into the above finding that the finite-size critical

temperature fluctuates according to the TW distribution.Let us consider a version of the PSM where spins Si arenot binary, but continuous variables satisfying the spher-ical constraint

PNi=1

S2

i = N [26]. This model is knownas the spherical PSM, and it has exactly the same be-havior as the PSM with Ising spins above [22], but itis more suitable for analytical studies [16]. Let us nowconsider this PSM for any finite p > 2 close to the transi-tion point: The system is in the high (low) temperaturephase if the average internal energy E is higher (lower)than the average energy Ec of the local energy minima[16]. Now, consider a finite-size sample J of the PSM.If the system’s internal energy E is larger than the en-ergy of the lowest local minimum, the system is in thehigh-temperature phase where it explores exponentiallymany local minima. Otherwise, the system is in the low-temperature phase where it explores the low-lying min-ima [16]. Hence, the critical energy EcJ of this sampleis given by the energy of the lowest local minimum, asillustrated in Fig. 4. Since the spin variables are contin-uous, we can introduce the Hessian matrix of the Hamil-tonian @2H/(@Si@Sj) evaluated in a local minimum, andits smallest eigenvalue �min. The fluctuations of EcJcan be understood by considering the geometry of thelocal energy minima. Indeed, following [11] the deeperthe minimum, the larger the curvature of H[

~S] vs. ~S atthe minimum, and so the larger �min, as illustrated inFig. 4: Let us then assume [11] that the energy EcJof the lowest-lying minimum is set by the smallest eigen-value �min of the Hessian in such minimum. Since for anyp > 2 the Hessian is a random matrix belonging to theGaussian orthogonal ensemble (GOE) [26, 27], its small-est eigenvalue �min is distributed according to the TWdistribution [17]. Given that �min is TW distributed, theabove geometrical argument shows that the fluctuationsof the critical energy EcJ , and consequently the fluctu-ations of the critical temperature �cJ , are described bythe TW distribution. Importantly, this argument showsthat the emergence of the TW distribution shown in thenumerical simulations for p = 3 holds for any finite p > 2.Finally, let us discuss the large-p limit: In this limit, thePSM is equivalent to the REM [12], and the distribu-tion of the finite-size critical temperature converges tothe Gumbel distribution. Indeed, for large p the abovelocal geometrical structure of the energy landscape (seeFig. 4) disappears, and the local minima become sim-ply a set of IID Gaussian random variables. Given thatthe energy threshold EcJ is the minimum of these IIDGaussian random variables, EcJ is distributed accord-ing to the Gumbel distribution [28], and so is the criticaltemperature.

Conclusions - In this Letter we have studied the finite-size fluctuations of the freezing-transition temperature oftwo mean-field models of structural glasses: The randomenergy model (REM) [12] and the p-spin model (PSM)[13] with p = 3. We find that for both the REM and thePSM, the finite-size fluctuations of the critical temper-ature are described by extreme-value-statistics probabil-ity distributions. For the REM, the critical-temperature

4

H[~S]

EcJ

I

II

III

�II

min

�I

min

�III

min

~S

FIG. 4: Schematic energy landscape of the spherical p-spinmodel: Energy H[~S] of a given sample J as a function ofthe spin configuration ~S. Three local energy minima labeledas I, II, III (left), and one low-lying minimum (right) are de-picted. The energy of minimum II coincides with the criticalenergy EcJ of the sample. The smallest eigenvalues of theHessian matrix @2H/(@Si@Sj) computed in each local mini-mum are �I

min,�IImin,�

IIImin. The geometry of the minima sets

the energy threshold: Since here �Imin < �III

min < �IImin, min-

imum II has the largest curvature, thus the lowest energyamongst the three local energy minima.

fluctuations are described by the Gumbel distribution[28], while for the PSM the critical temperature is dis-tributed according to the Tracy-Widom (TW) distribu-tion, which has been recently discovered in the theory ofrandom matrices [17]. For the PSM, we have provided ananalytical argument to understand the emergence of theTW distribution. Recent studies [7, 8] have shown thatthe TW distribution also emerges in spin glasses: Theabove analytical argument shows that the physics under-lying the emergence of the TW distribution in structuralglasses is completely different from spin glasses, becauseit involves a different mechanism related to the statisticsof glassy metastable states. Taken together, the resultsprovided in this Letter establish an unprecedented con-nection between the theory of extreme values and thefreezing transition of structural glasses.

As a topic of future studies, it would be interest-ing to study the fluctuations of the critical temperaturein structural-glass models with short-range interactions,such as the Edwards-Anderson (EA) model in an externalmagnetic field [29]. In this regard, previous studies haveshown that EVS distributions play a role in short-rangesystems with quenched disorder, such as the EA modelwith no external field [8]. In fact, one could imagineshort-range systems to behave as an ensemble of nearlyindependent subsystems, each subsystem having its owncritical temperature: The finite-size critical temperatureof the system as a whole is then given by the smallestof the subsystems’ critical temperatures [8]. This raisesthe possibility that the crtical temperature of the sys-tem as a whole could be distributed according one of thethree EVS distributions of independent and identicallydistributed random variables: The Gumbel, Fréchet, orWeibull distribution, as predicted by the extreme-valuetheorem [28].

Acknowledgments - We would like to thank C. Broed-ersz and G. Schehr for useful discussions. Researchsupported by NSF Grants No. PHYâĂŞ0957573 andCCFâĂŞ0939370, by the Human Frontiers Science Pro-gram, by the Swartz Foundation, and by the W. M.Keck Foundation. Simulations were performed on com-putational resources supported by the Office of Informa-tion Technology’s High Performance Computing Centerat Princeton University.

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5

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Supplemental Material for “Extreme-Value Distributions and the Freezing Transition

of Structural Glasses”

Michele Castellana⇤

Joseph Henry Laboratories of Physics and Lewis–Sigler Institute for Integrative Genomics,Princeton University, Princeton, New Jersey 08544

Convergence to the infinite-size critical temperature

In this Section we show that the finite-size critical temperature for the p-spin model defined in Eq. (1) convergesto the infinite-size critical temperature �

c

= 1.535 [1]. To do so, we compute the quantity ��

c

⌘ [(�cJ � �

c

)2]1/2,representing the deviation of the finite-size critical temperature �

cJ from the infinite-size critical temperature �

c

:The smaller ��

c

, the more peaked the distribution of �cJ around �

c

. Fig. S1 shows that ��

c

is a decreasing functionof the system size N : Hence, the numerical data suggest that for large N the finite-size critical temperature �

cJconverges to its infinite-size value �

c

.

0.1

0.15

0.2

0.25

16 32 64 128 256

��

c

N

FIG. S1: Deviation ��c of the finite-size critical temperature �cJ from the infinite-size critical temperature �c as a function

of the system size N .

Statistical-hypothesis test

In this Section we present a statistical test supporting the emergence of the Tracy-Widom (TW) distribution as thedistribution of the finite-size critical temperature of the p-spin model. First, in Fig. S2 we show p(xJ ) for multiplesystem sizes N with the relative error bars, which have been estimated with the bootstrap method [2]: This Figureshows how the distribution p(xJ ) compares to the TW distribution relatively to the numerical error bars. Second, thesamples of normalized finite-size critical temperatures xJ have been compared with the TW, Gumbel and Gaussiandistribution with zero mean and unit variance by using the Kolmogorov-Smirnov (KS) test [3]. The KS statistic isD = sup

x

|FJ (x) � F (x)|, where FJ is the cumulative distribution function (CDF) of xJ , F is the CDF of the TW,Gumbel or Gaussian distribution with zero mean and unit variance. The p-value is Pr(x �

pSD), where x is a random

variable distributed according to the Kolmogorov distribution [3] and S is the number of samples. Table S1 shows thep-value for system sizes N = 16, 32, 64, 128 shown in Fig. S1. At a significance level of 0.1, the data do not indicatethat the hypothesis that the finite-size critical temperature is distributed according to the TW distribution shouldbe rejected for N > 16, while at the same significance level the hypothesis that �

cJ is distributed according to theGaussian or Gumbel distribution is rejected for all the values of N shown in Table S1.

⇤Electronic address: michelec@princeton.edu

[1] A. Billoire, L. Giomi, and E. Marinari, Europhys. Lett. 71, 824 (2005).

[2] B. Efron, Ann. Stati. 7, 1 (1979).

[3] A. M. Mood, Introduction to the theory of statistics (McGraw-hill, 1950).

2

10

�4

10

�3

10

�2

10

�1

�4 �2 0 2 4

p(x

J)

xJ

10

�4

10

�3

10

�2

10

�1

�4 �2 0 2 4

p(x

J)

xJ

10

�4

10

�3

10

�2

10

�1

�4 �2 0 2 4

p(x

J)

xJ

10

�4

10

�3

10

�2

10

�1

�4 �2 0 2 4

p(x

J)

xJ

10

�4

10

�3

10

�2

10

�1

�4 �2 0 2 4

p(x

J)

xJ

10

�4

10

�3

10

�2

10

�1

�4 �2 0 2 4

p(x

J)

xJ

10

�4

10

�3

10

�2

10

�1

�4 �2 0 2 4

p(x

J)

xJ

10

�4

10

�3

10

�2

10

�1

�4 �2 0 2 4

p(x

J)

xJ

0

0.2

0.4

�2 0 2

p(x

J)

xJ

0

0.2

0.4

�2 0 2

p(x

J)

xJ

0

0.2

0.4

�2 0 2

p(x

J)

xJ

0

0.2

0.4

�2 0 2

p(x

J)

xJ

0

0.2

0.4

�2 0 2

p(x

J)

xJ

0

0.2

0.4

�2 0 2

p(x

J)

xJ

0

0.2

0.4

�2 0 2

p(x

J)

xJ

0

0.2

0.4

�2 0 2

p(x

J)

xJ

Tracy-Widom

Gumbel

Gaussian

N = 64 Tracy-Widom

Gumbel

Gaussian

N = 128

Tracy-Widom

Gumbel

Gaussian

N = 32Tracy-Widom

Gumbel

Gaussian

N = 16

FIG. S2: Distribution p(xJ ) of the normalized critical temperature xJ with error bars, for di↵erent system sizes N =

16, 32, 64, 128 (in red, blue, brown, black respectively). Each system size is represented in a di↵erent panel, together with the

Tracy-Widom, Gumbel and Gaussian distribution, all with zero mean and unit variance. Insets: Same plots as in the main

panels in linear scale.

N S pTW pGumbel pGaussian

16 1.3⇥ 10

51.1⇥ 10

�5

< 10

�10< 10

�1032 1.3⇥ 10

50.26

64 1.3⇥ 10

50.25

128 6.9⇥ 10

40.88

TABLE S1: Kolmogorov-Smirnov (KS) test for the distribution of the finite-size critical temperature: For system sizes

N = 16, 32, 64, 128 we show the number of samples S and the p-value of the KS test which compares the distribution of the

normalized critical temperature with a reference probability distribution: The Tracy-Widom, Gaussian or Gumbel distribution,

all with zero mean and unit variance. Small value of the p-value suggest that the distribution of the normalized critical

temperature is significantly di↵erent from the reference distribution.