Free energy landscapes, dynamics and the edge of chaos in mean-field models of spin glasses

A. J. Bray; M. A. Moore; R. A. Blythe; T. Aspelmeier

REVIEW

Free energy landscapes, dynamics and the edge of chaos in mean-field models of spin glasses

T. Aspelmeier, R. A. Blythe, A. J. Bray, M. A. Moore

Disordered Systems and Neural Networks · Submitted 27 Feb 2006 · 23 references

Pith's verdict: The paired-minima picture and the edge-of-chaos algorithm are solid, but the claimed B ~ (f-f0)^(-1/3) barrier law is extracted from a biased sample and needs independent confirmation before it becomes a landscape result.

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arXiv IDcond-mat/0602639 (version 1)

Pith IDY6IFMUBT, Pith's permanent record of this paper and its review

CitationA. J. Bray, M. A. Moore, R. A. Blythe, T. Aspelmeier. "Free energy landscapes, dynamics and the edge of chaos in mean-field models of spin glasses." arXiv:cond-mat/0602639 (2006). https://pith.science/paper/Y6IFMUBT

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Editor's note

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The useful headline: this paper gives the clearest numerical picture yet of TAP versus NMF free-energy landscapes, and it introduces an iterative scheme that actually finds very low free-energy states. The paired minima/index-1 saddles in TAP are confirmed, and the NMF saddle complexity scaling in K/N is a genuinely new and reasonably solid result. The edge-of-chaos algorithm is creative and demonstrably competitive, so the paper deserves a serious referee.

The soft spot is the load-bearing barrier claim. The paper states B ~ (f-f0)^(-1/3), which leads to N^(1/3) barriers for pure states. But that relation is obtained entirely from solutions produced by the alpha-controlled iteration, and the authors themselves concede in V.F that "the states of a given free energy reached by our particular algorithm almost certainly do not sample all states of that free energy uniformly." If the algorithm preferentially finds low-lying states surrounded by high barriers (which the same section argues is the case), then the observed correlation in Fig. 12 could be a property of the sampling, not the landscape. This is not a fatal flaw, but it is the central claim, and the paper's own caveat makes it unavoidable.

Two more worries, in order. The finite-size scaling for barrier heights (Fig. 11) is admitted to be "not very good," and the free-energy exponent gamma is stated as roughly 1 with no error bar. The pure-state regime f-f0 ~ 1/N is exactly where success rates vanish exponentially, so no direct data at large N tests the N^(1/3) law. The collapse in Fig. 12 is impressive across a wide range of N and alpha, which is evidence against an easy artifact, but it is still a collapse of algorithm-selected states.

What the paper does well: the exhaustive NMF enumeration with Morse-sum checks, the Hessian eigenvector overlaps supporting the paired picture, and the careful treatment of numerical pathologies (transformed variables, arbitrary precision) are all honest and careful work. The NMF complexity curves collapse nicely, and the contrast between the robust NMF landscape and the fragile TAP landscape is well argued.

Bottom line: cite the NMF complexity result and the edge-of-chaos heuristic; cite the barrier scaling only with a caveat. Send to referee, but ask for an independent test of the barrier law — for instance, exhaustive enumeration of all TAP turning points at moderate N, or a different algorithm that does not tune alpha to the edge of chaos. The paper is good enough to warrant that effort.

Referee report

Significance

If the barrier scaling law is established, the paper would have a significant consequence: most TAP metastable states would be separated by finite, small barriers, whereas only the lowest free-energy states would have barriers growing as N^(1/3). This would sharpen the contrast between TAP and NMF landscapes and bear on the physical status of pure states in the SK model. The numerical work has notable strengths: the NMF complexity collapse in Fig. 2 is convincing; the Hessian spectral analysis in Figs. 13-15, performed with arbitrary-precision diagonalization, provides independent-looking support for the paired minimum-saddle structure; and the free-energy and success-rate scaling studies (Figs. 7 and 10) are systematic. However, the central barrier relation is extracted from an ensemble produced by a deliberately biased algorithm, and the paper itself concedes non-uniform sampling in Section V.F. The claim that B ~ (f-f0)^(-1/3) is a property of the whole TAP landscape therefore needs substantially stronger support before the N^(1/3) pure-state conclusion can be accepted.

Major objections (3)

[Section V.F, after Fig. 11] The paper concedes that 'the states of a given free energy reached by our particular algorithm almost certainly do not sample all states of that free energy uniformly.' Since the central relation B ~ (f-f0)^(-1/3) (Fig. 12) is measured only on the sample produced by the alpha-controlled iteration, it is not established that the relation is a property of the full TAP landscape rather than a selection effect. The algorithm may preferentially return low-free-energy states whose surrounding barriers are unusually high, and this could generate the observed correlation between B and f-f0. Please provide an unbiased check, such as exhaustive enumeration of all minimum-saddle pairs for small N (as done for the NMF equations in Section III) or sampling with an independent solver, and demonstrate that the barrier relation is insensitive to the sampling mechanism.
[Section V.F, Eq. (13), Figs. 7, 10, 11] The scaling exponent gamma is quoted as approximately 1 'with too much scatter to provide an error bar,' and the barrier scaling collapse in Fig. 11 is admitted to be 'not very good.' Moreover, the pure-state regime x = N(alpha-alpha_c) << 1 is exactly the regime where success rates vanish as exp(-const. N^(4 zeta / 3)) according to Section V.E and Fig. 10, so no direct data constrain the N^(1/3) law at f-f0 ~ 1/N. The central claim therefore rests on extrapolating a poorly collapsing scaling plot into a regime where the algorithm finds essentially no states. The paper should either provide a substantially better collapse with error estimates, supply data at fixed small f-f0 obtained by other means, or give an analytic derivation of the exponent and coefficient.
[Section V.F, Fig. 12] In Fig. 12, each data point is the average over all solutions found with a particular value of alpha, so the parameter alpha controls both plotted coordinates. Because the sampled subset of states changes as alpha is varied, the statement that the alpha dependence is eliminated does not rule out the possibility that the apparent B ~ (f-f0)^(-1/3) law is generated by the algorithm's selection function. To make the landscape interpretation convincing, the authors should show the same relation for a set of states not selected by the alpha-tuned map, for example all TAP pairs in small systems or states obtained from a qualitatively different solver, and report the scatter across such independent samples.

Minor objections (6)

[Abstract and title] The abstract contains 'NME equations' where the text otherwise uses NMF, and the title has a typo: 'mea n-ﬁeld' should be 'mean-ﬁeld.'
[Section II.C] The text references 'Fig. II C' twice; this should be 'Fig. 1' to match the figure numbering.
[Table II] Table II uses commas as decimal separators in entries such as '0,048' and '0,02', which is inconsistent with the decimal-point notation used elsewhere in the paper.
[Section VI] The final paragraph begins 'We conclude with with some general remarks'; the duplicated 'with' should be removed.
[Figures 7, 9, 10, 11, 12] Several scaling plots would benefit from explicit error bars and a statement of the number of disorder samples and starting points used at each N; this is particularly important for Fig. 12, where the claimed master relation is central but the sampling statistics are not reported.
[Section V.E] The Lifshitz exponent zeta is quoted as approximately 2 without an error estimate; since the success-rate scaling is used to argue that the pure-state regime is inaccessible, a quantitative uncertainty would help the reader judge the strength of that argument.

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The paper

The paper claims

The barrier between a spin-glass minimum and its paired saddle scales as $(f-f_0)^{-1/3}$, so pure-state barriers grow as $N^{1/3}$.

The reading

This paper studies the free-energy landscapes behind two mean-field descriptions of Ising spin glasses. For the TAP equations, it claims the landscape contains only two kinds of stationary points—local minima and index-one saddles—and that each minimum is paired with a nearby saddle. The free-energy difference between a minimum and its partner saddle, the barrier, is claimed to scale as $(f-f_0)^{-1/3}$, where $f$ is the free energy per spin of the solution and $f_0$ is the equilibrium value. For pure states, where $f-f_0 \sim 1/N$, this gives barriers growing as $N^{1/3}$; for states far above equilibrium the barriers stay finite and small. The paper also introduces a one-parameter iterative algorithm tuned to the edge of chaos that can reach near-equilibrium states, while the naive-mean-field landscape is shown to have many high-index saddles and a threshold free energy that traps descent dynamics.

What carries the argument

The load-bearing object is the generalized mean-field free energy $F = -\sum_{(i,j)} J_{ij} m_i m_j - \gamma\beta N(1-q)^2/4 + \beta^{-1}\sum_i [\frac{1+m_i}{2}\ln\frac{1+m_i}{2} + \frac{1-m_i}{2}\ln\frac{1-m_i}{2}]$, whose stationary points solve $m_i = \tanh[\beta h_i - \gamma\beta^2(1-q)m_i]$, with $\gamma=1$ for TAP and $\gamma=0$ for naive mean field. The Hessian (matrix of second derivatives) classifies each solution by its index, the number of negative eigenvalues, and the central structural claim is that for TAP only index 0 and index 1 occur, in paired clusters. The second piece of machinery is the modified iteration $m_i^{(k+1)} = m_i^{(k)} + \alpha[G_i(m^{(k)}) - m_i^{(k)}]$; tuning $\alpha$ to its critical value, the edge of chaos, controls which solutions are found and yields the finite-size scaling used to extract the barrier relation.

Extended reading notes

Core claim

The central discovery is that the TAP free-energy landscape of the infinite-range Ising spin glass is structurally simple: every nontrivial solution is either a minimum or an index-one saddle (exactly one downward direction), and minima and saddles come in close pairs. For each pair, the barrier height $B$ obeys $B \sim (f-f_0)^{-1/3}$, with no separate dependence on system size or the algorithm parameter, so the barrier is a property of the landscape itself. Since pure states have $f-f_0 = O(1/N)$, their barriers grow as $N^{1/3}$, whereas TAP states with $f-f_0 = O(1)$ have finite, small barriers and are therefore dynamically inconsequential. The paper further claims that the naive-mean-field equations, identical at zero temperature, produce a qualitatively different landscape with saddles of all indices up to $O(N)$, a complexity that scales with $K/N$, and a threshold below which minima exponentially outnumber saddles; descent dynamics on that surface gets trapped near this threshold.

Load-bearing premise

The results assume the states reached by the tuned iteration fairly represent all TAP minimum-saddle pairs at a given free energy; the paper admits that its algorithm almost certainly does not sample all states of that free energy uniformly, so the barrier scaling could in principle be an artifact of which pairs the algorithm finds.

Editorial extensions

If this is right

If the barrier relation holds, the lowest-lying TAP states, the pure states, become increasingly hard to escape as $N$ grows, with escape barriers of order $N^{1/3}$.
Most TAP metastable states, those with $f-f_0$ of order one, have barriers that do not grow with $N$, so they should not act as long-lived traps in the dynamics of the SK model.
The edge-of-chaos iteration can in principle approach the equilibrium free energy per spin, but the success probability falls faster than a power of $N$, consistent with a Lifshitz-like tail, so finding near-equilibrium states becomes drastically harder as $N$ grows.
For the NMF equations, the free energy at which minima outnumber saddles acts as a threshold below which descent-type dynamics cannot go; this explains why different iterative algorithms all converge to nearly the same free energy.
A practical consequence: when constructing finite-temperature approximations to an optimization problem, a 'fragile' landscape like TAP with paired low-index saddles is preferable to a rugged high-index one like NMF.
If pure-state barriers scale as $N^{1/3}$, activated escape times should scale as $\exp(c N^{1/3})$, putting SK dynamics in a regime between Arrhenius (order-one barrier) and exponentially hard ($N^\alpha$) traps; this could be tested in off-equilibrium dynamics simulations.
The absence of high-index saddles and the pairing of minima with index-one saddles suggests the TAP landscape may be gradient-like: descent from any point could reach a minimum without crossing saddles of index greater than one. If true, TAP-based optimization would avoid the trapping seen on rugged landscapes, a property worth testing on other mean-field systems.
The edge-of-chaos algorithm's excursions outside the physical hypercube ($|m_i|>1$) are an intentional feature that lets the iteration tunnel through barriers; this suggests a general heuristic for hard optimization, but the rapid decay of success probability with $N$ cautions that practical gains may be limited to moderate sizes unless the control parameter is adaptively scheduled during a run.

Original abstract

Metastable states in Ising spin-glass models are studied by finding iterative solutions of mean-field equations for the local magnetizations. Two different equations are studied: the TAP equations which are exact for the SK model, and the simpler `naive-mean-field' (NMF) equations. The free-energy landscapes that emerge are very different. For the TAP equations, the numerical studies confirm the analytical results of Aspelmeier et al., which predict that TAP states consist of close pairs of minima and index-one (one unstable direction) saddle points, while for the NMF equations saddle points with large indices are found. For TAP the barrier height between a minimum and its nearby saddle point scales as (f-f_0)^{-1/3} where f is the free energy per spin of the solution and f_0 is the equilibrium free energy per spin. This means that for `pure states', for which f-f_0 is of order 1/N, the barriers scale as N^{1/3}, but between states for which f-f_0 is of order one the barriers are finite and also small so such metastable states will be of limited physical significance. For the NMF equations there are saddles of index K and we can demonstrate that their complexity Sigma_K scales as a function of K/N. We have also employed an iterative scheme with a free parameter that can be adjusted to bring the system of equations close to the `edge of chaos'. Both for the TAP and NME equations it is possible with this approach to find metastable states whose free energy per spin is close to f_0. As N increases, it becomes harder and harder to find solutions near the edge of chaos, but nevertheless the results which can be obtained are competitive with those achieved by more time-consuming computing methods and suggest that this method may be of general utility.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.89 · human participants 0.01 · controlled comparison 0.15 · randomized assignment 0.01 · states quantitative result 0.93 · compares with prior results 0.95 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Metastable states in Ising spin-glass models are studied by finding iterative solutions of mean-field equations for the local magnetizations. Two different equations are studied: the TAP equations which are exact for the SK model, and the simpler `naive-mean-field' (NMF) equations. The free-energy landscapes that emerge are very different. For the TAP equations, the numerical studies confirm the analytical results of Aspelmeier et al., which predict that TAP states consist of close pairs of minima and index-one (one unstable direction) saddle points, while for the NMF equations saddle points with large indices are found. For TAP the barrier height between a minimum and its nearby saddle point scales as (f-f_0)^{-1/3} where f is the free energy per spin of the solution and f_0 is the equilibrium free energy per spin. This means that for `pure states', for which f-f_0 is of order 1/N, the barriers scale as N^{1/3}, but between states for which f-f_0 is of order one the barriers are finite and also small so such metastable states will be of limited physical significance. For the NMF equations there are saddles of index K and we can demonstrate that their complexity Sigma_K scales as a function of K/N. We have also employed an iterative scheme with a free parameter that can be adjusted to bring the system of equations close to the `edge of chaos'. Both for the TAP and NME equations it is possible with this approach to find metastable states whose free energy per spin is close to f_0. As N increases, it becomes harder and harder to find solutions near the edge of chaos, but nevertheless the results which can be obtained are competitive with those achieved by more time-consuming computing methods and suggest that this method may be of general utility.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Metastable states in Ising spin-glass models are studied by finding iterative solutions of mean-field equations for the local magnetizations.	method 0.74	0.74	0.03	asserted 0.99
s1	Two different equations are studied: the TAP equations which are exact for the SK model, and the simpler `naive-mean-field' (NMF) equations.	method 0.92	0.50	0.03	asserted 0.97
s2	The free-energy landscapes that emerge are very different.	result 0.97	0.69	0.08	asserted 1.00
s3	For the TAP equations, the numerical studies confirm the analytical results of Aspelmeier et al., which predict that TAP states consist of close pairs of minima and index-one (one unstable direction) saddle points, while for the NMF equations saddle points with large indices are found.	result 1.00	0.73	0.23	asserted 1.00
s4	For TAP the barrier height between a minimum and its nearby saddle point scales as (f-f_0)^{-1/3} where f is the free energy per spin of the solution and f_0 is the equilibrium free energy per spin.	result 1.00	0.63	0.10	asserted 1.00
s5	This means that for `pure states', for which f-f_0 is of order 1/N, the barriers scale as N^{1/3}, but between states for which f-f_0 is of order one the barriers are finite and also small so such metastable states will be of limited physical significance.	interpretation 0.98	0.66	0.15	asserted 0.92
s6	For the NMF equations there are saddles of index K and we can demonstrate that their complexity Sigma_K scales as a function of K/N.	result 0.99	0.91	0.09	asserted 1.00
s7	We have also employed an iterative scheme with a free parameter that can be adjusted to bring the system of equations close to the `edge of chaos'.	method 0.93	0.95	0.06	asserted 0.95
s8	Both for the TAP and NME equations it is possible with this approach to find metastable states whose free energy per spin is close to f_0.	result 0.98	0.92	0.14	hedged possibility 0.81
s9	As N increases, it becomes harder and harder to find solutions near the edge of chaos, but nevertheless the results which can be obtained are competitive with those achieved by more time-consuming computing methods and suggest that this method may be of general utility.	interpretation 0.73	0.89	0.32	hedged possibility 0.92

What kind of paper the classifier reports this is: simulation study 0.94 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1: Energies reached from solutions of the NMF equations

**Figure 4.** FIG. 4: Eigenvalue distributions of the Hessian at minima,

**Figure 7.** FIG. 7: Finite-size scaling of the free energy as a function o

**Figure 9.** FIG. 9: Success rates (probability of finding a solution) as a

**Figure 10.** FIG. 10: Success rates as a function of

**Figure 12.** FIG. 12: The (total) barrier height as a function of free ener

**Figure 13.** FIG. 13: Finite size scaling plot of the average smallest eig

**Figure 15.** FIG. 15: Plot of the difference between the second and the

**Figure 16.** FIG. 16: Finite-size scaling of the NMF free-energy at

Showing 12 of the paper's 16 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:cond-mat/0602639 by the authors.

References

Works cited (23)

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F. Tanaka and S. F. Edwards, J. Phys. F 10, 2769 (1980)

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[2]

A. J. Bray and M. A. Moore, J. Phys.C 13, L469 (1980)

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D. J. Thouless, P. W. Anderson, and R. G. Palmer, Phil. Mag. 35, 593 (1977)

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G. Parisi and T. Rizzo, J. Phys. A 37, 7979 and references therein (2004)

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T. Aspelmeier, A. J. Bray, and M. A. Moore, Phys. Rev. Lett. 92, 087203 (2004)

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[6]

A. Cavagna, I. Giardina, and G. Parisi, Phys. Rev. Lett. 92, 120603 (2004)

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T. Castellani and A. Cavagna, cond-mat/0505032 and references therein (2005)

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S. Boettcher and M. Frank, cond-mat/0509001 (2005)

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A. J. Bray, H. Sompolinsky, and C. Yu, J. Phys. C: Solid State Phys. 19, 6389 (1986)

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[10]

M. M´ ezard, G. Parisi, and M. A. Virasoro, Spin glass theory and beyond (World Scientiﬁc, Singapore, 1987)

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T. Plefka, Phys. Rev. B 65, 224206 (2002)

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C. G. Broyden, Math. Comput. 19, 577 (1965)

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The GNU scientiﬁc library (GSL), version 1.4 , URL http://www.gnu.org/software/gsl

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T. S. Grigera, A. Cavagna, I. Giardina, and G. Parisi, Phys. Rev. Lett 88, 055502 (2002)

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W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. Vet- terling, Numerical recipes in C (Cambridge University Press, Cambridge, 1992)

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D. H. Bailey, Y. Hida, K. Jeyabalan, X. S. Li, and B. Thompson, Arbitrary pre- cision package (ARPREC) (2005), URL http://crd.lbl.gov/~dhbailey/mpdist/index.html

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A. Crisanti and T. Rizzo, Phys. Rev. E 65, 046137 (2002)

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A. Crisanti, L. Leuzzi, G. Parisi, and T. Rizzo, Phys. Rev. B 68, 174401 (2003)

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[19]

T. Rizzo, private communication

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[20]

at all. There is another very distinct change between α = 0.99 and α = 0.95 from inﬁnite limit cycles to cycles of length 2. Closer inspection revealed that these latter cycles are of the form mi → −mi →mi. The picture which emerges from this data is that for α > 1 we ﬁnd a distribution of cycle lengths which in- cludes a ﬁnite fraction of cycles with len...

View on Pith
[21]

G. J. Rodgers and M. A. Moore, J. Phys. A 22, 1085 (1989)

View on Pith 1989
[22]

C. M. Newman and D. L. Stein, Phys. Rev. E 60, 5244 (1999)

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[40]

The data presented in the following are obtained by amalgamating results from all samples, including those whose solution sets were certainly incomplete by virtue of SM ⁄= 1. Excluding these samples from the analysis changes our results very little. B. Results We ﬁrst examine what classes of turning point arise in the free-energy landscape. In the TAP fre...

View on Pith

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

F. Tanaka and S. F. Edwards, J. Phys. F 10, 2769 (1980)

View on Pith 1980

[2] [2]

A. J. Bray and M. A. Moore, J. Phys.C 13, L469 (1980)

View on Pith 1980

[3] [3]

D. J. Thouless, P. W. Anderson, and R. G. Palmer, Phil. Mag. 35, 593 (1977)

View on Pith 1977

[4] [4]

G. Parisi and T. Rizzo, J. Phys. A 37, 7979 and references therein (2004)

View on Pith 2004

[5] [5]

T. Aspelmeier, A. J. Bray, and M. A. Moore, Phys. Rev. Lett. 92, 087203 (2004)

View on Pith 2004

[6] [6]

A. Cavagna, I. Giardina, and G. Parisi, Phys. Rev. Lett. 92, 120603 (2004)

View on Pith 2004

[7] [7]

T. Castellani and A. Cavagna, cond-mat/0505032 and references therein (2005)

View on Pith Pith review arXiv 2005

[8] [8]

S. Boettcher and M. Frank, cond-mat/0509001 (2005)

View on Pith Pith review arXiv 2005

[9] [9]

A. J. Bray, H. Sompolinsky, and C. Yu, J. Phys. C: Solid State Phys. 19, 6389 (1986)

View on Pith 1986

[10] [10]

M. M´ ezard, G. Parisi, and M. A. Virasoro, Spin glass theory and beyond (World Scientiﬁc, Singapore, 1987)

View on Pith 1987

[11] [11]

T. Plefka, Phys. Rev. B 65, 224206 (2002)

View on Pith 2002

[12] [12]

C. G. Broyden, Math. Comput. 19, 577 (1965)

View on Pith 1965

[13] [13]

The GNU scientiﬁc library (GSL), version 1.4 , URL http://www.gnu.org/software/gsl

View on Pith

[14] [14]

T. S. Grigera, A. Cavagna, I. Giardina, and G. Parisi, Phys. Rev. Lett 88, 055502 (2002)

View on Pith 2002

[15] [15]

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. Vet- terling, Numerical recipes in C (Cambridge University Press, Cambridge, 1992)

View on Pith 1992

[16] [16]

D. H. Bailey, Y. Hida, K. Jeyabalan, X. S. Li, and B. Thompson, Arbitrary pre- cision package (ARPREC) (2005), URL http://crd.lbl.gov/~dhbailey/mpdist/index.html

View on Pith 2005

[17] [17]

A. Crisanti and T. Rizzo, Phys. Rev. E 65, 046137 (2002)

View on Pith 2002

[18] [18]

A. Crisanti, L. Leuzzi, G. Parisi, and T. Rizzo, Phys. Rev. B 68, 174401 (2003)

View on Pith 2003

[19] [19]

T. Rizzo, private communication

View on Pith

[20] [20]

at all. There is another very distinct change between α = 0.99 and α = 0.95 from inﬁnite limit cycles to cycles of length 2. Closer inspection revealed that these latter cycles are of the form mi → −mi →mi. The picture which emerges from this data is that for α > 1 we ﬁnd a distribution of cycle lengths which in- cludes a ﬁnite fraction of cycles with len...

View on Pith

[21] [21]

G. J. Rodgers and M. A. Moore, J. Phys. A 22, 1085 (1989)

View on Pith 1989

[22] [22]

C. M. Newman and D. L. Stein, Phys. Rev. E 60, 5244 (1999)

View on Pith 1999

[23] [40]

The data presented in the following are obtained by amalgamating results from all samples, including those whose solution sets were certainly incomplete by virtue of SM ⁄= 1. Excluding these samples from the analysis changes our results very little. B. Results We ﬁrst examine what classes of turning point arise in the free-energy landscape. In the TAP fre...

View on Pith