The runs choose a mode near β = 1, where the field’s push toward A is still weak next to random fluctuations. After the modes form, switching can take much longer than settling within one (paper Sec. 3).

The dashed mark on the c meter is the selection formula (Eq. 3) for this field, size and cooling rate, divided by the model’s own log-odds; nothing is fitted. Try more sweeps per run: slower cooling helps, but less than you might expect (chapter 5).

The formula and the headline’s 0.199 are linked by one step. On linear ramps the formula matches the exact CW chain within 1.81%, with the chain slightly above it in all nine cases (Table 2). On the paper’s cooling schedule, that chain gives c_CW = 0.120, and the amplification factor A₃ = 1.662 for three patterns turns it into 0.199 (chapter 6).

No. Fig. 0 runs a small two-mode model (N = 256, a linear ramp) so that it runs live in your browser, and it has its own c. Its closeness to 0.199 is a coincidence of the settings: c changes with model size and schedule.

The headline above is the paper’s test (chapter 7): three stored patterns at N = 2,048, predicted 0.120 × 1.662 = 0.199 (nothing fitted to the network runs), measured 0.199 (95%: 0.1908–0.2095). Besides this test, p = 4 also falls inside its interval; the other settings miss by up to 11% (Limits 1).

Near β = 1 a small random imbalance between + and − spins can grow into either mode, and the field biases this early choice. Once the modes have formed, switching between them can take much longer than settling within one (Levin, Luczak and Peres 2010; Ding, Lubetzky and Peres 2009), so the sign chosen early can decide the final proportions.

The rest of the page asks how much of a small preference survives that choice.

The paper uses the simplest model with two modes, the Curie–Weiss model, whose probabilities and sampler both reduce exactly to the count of + spins (Sec. 2): N binary spins that all pull toward agreeing with each other,

G is the field, fixed during a cooling run. Mode A is “mostly +1”, mode B is “mostly −1”, and the mean spin, the magnetisation m, says which one a sample is in. For large N the magnetisation approximately follows the drift (App. A)

With no field, for β < 1 only m = 0 is stable; for β > 1 two stable states appear at opposite nonzero m.

The Gibbs sampler redraws one spin at a time. At the transition the landscape over m has a flat bottom: the linear pull back to equilibrium is zero, and only the weak cubic pull m³/3 remains. Physicists call this critical slowing down. This chapter is a side note: you can skip to chapter 3.

The relaxation time τ is the inverse spectral gap: the time for the slowest deviation from equilibrium to shrink by a factor e. For β ≤ 1 the schedule uses the full chain; above 1 it uses the chain held on one sign (reflected at m = 0), so τ measures settling within a mode. The dwell at each temperature is ⌈6τ⌉ sweeps, computed at N = 2,048 and reused at every size (Table 3). This is a relaxation rule, not a bound on the error in mode probabilities (App. A).

Redesigned for each N, a schedule that contains β = 1 would need a √N dwell at that temperature; here it is designed once and held fixed.

Near β = 1 the drift is approximately

At β = 1 only the cubic term is left, so in this large-N drift (the equation above: no field, no random kicks) the distance to rest decays as t^−1/2 instead of exponentially (set β = 1 above: dm/dt ≈ −m³/3 gives m ≈ √(3/(2t)) at late times; panel a integrates the full drift live).

A small field G pushes every spin toward +1, so the model puts more probability on mode A, the positive mode (m > 0); mode B is the negative one. A perfect sampler would produce mode A exactly as often as the model says. The figure counts what the paper’s sampler produces instead, and turns the gap into one number, the retained share c, measured on the log-odds scale.

q_eq is the model’s probability of mode A at the end of cooling and q_chain the share of runs that end there, both among samples with m ≠ 0. For a weak field the sampled and the model’s log-odds are both close to proportional to G (Sec. 3), so G nearly cancels in their ratio c. c = 1 means the sampler reproduces the model; c = 0 is a coin flip. If the model’s odds are 4:1, c = 0.2 gives about 1.32:1 (paper Sec. 2).

Where	Setting	c
Overview, The paper	two modes, N = 256, linear ramp of 250 sweeps	0.19 formula, computed here1
Chapter 4, Field against fluctuations	two modes, N = 1,024, linear ramp, γ = 0.001 per sweep (slider start)	0.13 formula, computed here
Chapter 5, Fig. 5a	two modes, N = 2,048, linear ramp of 225 sweeps	0.066 exact CW chain, computed here
Chapters 3 and 5b	two modes, N = 2,048, the paper’s protocol (brief cooling, a filtered start that keeps only the runs ending in mode B, reheat, then the final cooling over 834 sweeps; why: Where the runs start): the reference cCW (CW: the Curie–Weiss model)	0.120 exact CW chain
Chapters 6 and 7	three patterns (Hopfield model), N = 2,048, the paper’s protocol	0.199 predicted, 0.199 measured (paper Fig. 2b slope; Fig. 7b on this page)2

Larger models and faster schedules keep less, so each setting has its own c. Only the two shaded rows enter the test. At other pattern counts and sizes the prediction misses by up to 11% (chapter 7).

¹ The live count in that picture lands within a few hundredths of the formula.
² Based on the one-pattern Hopfield ratio (×1.063) instead, the prediction is 0.212, outside the interval (paper Sec. 5; chapter 7).

A natural guess is that the loss comes from too few sweeps at each value of β. Before the interval the field has nothing to choose between; after it, switching modes takes much longer than settling, so extra sweeps change little. Only very long sampling at the final temperature recovers the model’s proportions, but at N = 2,048 the zero-field mean time for a mode switch is about 5 × 10⁷ expected sweeps at β = 1.1 (Sec. 3, App. A).

Inside the interval the weak field competes with the fluctuations, so it tilts a coin rather than deciding it.

A classical calculation (Kondepudi and Nelson 1983; Kondepudi and Gao 1987) gives the chance of the positive sign, the selection formula (paper Eq. 3), from three inputs: the field G, the model size N, and the cooling rate γ:

It equals q_chain if later switching is negligible. It assumes a slow ramp starting well below β = 1; Nγ ≫ 1 is a sufficient small-noise condition (Sec. 3). It does not extend to arbitrarily slow cooling at fixed N.

Near β = 1 the mean spin follows the drift-plus-noise equation (paper Eq. 7), with β = 1 + γt and t in sweeps:

The growth of m becomes appreciable over about sweeps around β = 1. In that interval the field pushes m by about , while fluctuations of size about per sweep accumulate to about . A run lands on the A side with probability Φ(push ÷ spread), Φ the Gaussian CDF; push ÷ spread is a in (4.1).

Chapter 5 turns this ratio into the scaling law for c, (5.1), and tests it.

A slower ramp gives the field more time to push the chain toward the favoured mode, but it also gives the random fluctuations more time to spread. The push grows faster than the spread, but only by the fourth root of the time: 16× the sweeps buys 2× c (the math).

Fig. 5a checks this on a linear ramp; Fig. 5b runs the paper’s own schedule, which gives the reference c_CW = 0.120 that the final test builds on.

The sampler’s log-odds. G is the field per spin toward the positive mode. From the race of chapter 4, the chain’s log-odds grow as the field’s push over the fluctuation spread in the selection interval:

The model’s log-odds. The two modes differ in log-probability by an amount proportional to N G: a push of G on each of N spins. The chain’s log-odds grow as G√N γ^−1/4, the model’s as N G. Their ratio is

So bigger models lose more (4× the model size halves c), and slowing the whole schedule helps only as γ^−1/4: sixteen times the sweeps doubles c. This holds when the field is weak: a = G√N(π/γ)^1/4 ≪ 1. The exact CW chain agrees with the full formula, Eq. 3, for N = 1,024–4,096 and γ = 0.00025–0.004: the chain lies above the formula in all nine cases, by 0.23–1.81% (Table 2). The law does not extend to arbitrarily slow cooling at fixed N: long sampling at the final temperature recovers equilibrium, but at N = 2,048 the zero-field mean time for a mode switch is about 5 × 10⁷ expected sweeps at β = 1.1 (Sec. 3, App. A). On the paper’s cooling schedule, a quarter of the final cooling gives c = 0.099 and four times gives 0.147: 1.48× for sixteen times the final cooling (Sec. 4, App. A).

Fig. 5a · the rings and the line

The line is the selection formula (Kondepudi & Nelson 1983; Kondepudi & Gao 1987), computed on this page from q = Φ(G√N (π/γ)^1/4) (paper Eq. 3). Each ring is the continuous-time Gibbs dynamics, reduced exactly to N + 1 counts (one time unit = one expected sweep: N updates on average), under a linear schedule from β = 0.7 to 1.6 at rate γ. The 1× budget is γ = 0.004, a ramp of 225 sweeps.

The field is scaled with size (G·N = 1.254), so the model’s own odds are nearly the same at every N: the positive mode has probability 0.973, log-odds 3.57. So the readout’s c is the sampler’s log-odds ÷ 3.57. Over the nine runs, the exact chain lies above the formula by 0.23–1.81% (Table 2), over N²γ = 262 to 67,109.

Fig. 5b · the paper’s cooling schedule

The paper does not use a linear ramp. Its cooling schedule holds β at ten inverse temperatures from 0.6 to 1.6 and dwells ⌈6τ⌉ sweeps at each, where τ is the relaxation time (Table 3), the e-fold time of the slowest deviation from equilibrium. τ peaks at the transition β = 1, so the schedule spends most of its budget near it: β = 1 alone takes 355 of the 834 sweeps (43%), and β = 0.95 to 1.05 take 73%. The closed-form law assumes β rises at a constant rate; this schedule dwells at each temperature, longest at the transition, so here the exact CW chain is used instead.

The paper first runs a brief cooling (2 sweeps per temperature) and keeps the runs that ended in the negative mode (the filtered start). It reheats them at β = 0.7 for 30 sweeps, then the final cooling follows this schedule. The scale toggle multiplies only the sweeps of the final cooling. The field is G = 6.125×10⁻⁴, and the chain is the continuous-time Gibbs dynamics, reduced exactly to 2,049 counts. Every number comes from the exact CW chain at N = 2,048 unless marked.

Budget and size scans

Sixteen times the final cooling (×¼ → ×4) raises c from 0.099 to 0.147, 1.48× (Sec. 4, App. A), less than the 2× of the ramp law. From the repository’s computations, not in the paper: ×8 gives 0.174 (results/experiments/cw_scaling.json). A linear ramp from β = 0.7 to 1.6 in 834 sweeps gives 0.091, computed on this page from Eq. 3.

The size scan keeps the same 834 sweeps: each temperature keeps its N = 2,048 sweep count at every N, and the field is rescaled so the model’s odds stay nearly fixed (repository). Between N = 1,024 and 4,096 it falls as about N^−0.47. The paper’s Table 1 instead keeps |G| = 0.0006125 at every N, with c_CW = 0.171, 0.120, 0.090 at N = 1,024, 2,048, 4,096: N^−0.46 (two-point slope). The linear-ramp law gives −1/2, because γ stays fixed and so N²γ grows as N². The exponent for sampling the three-pattern Hopfield model (chapter 6) is about −0.36 (paper Sec. 5). From the repository’s size scan, not in the paper (results/experiments/cw_scaling.json): c falls from 0.229 at N = 512 to 0.071 at N = 8,192.

The one-pattern ratio

Every number here comes from the exact CW chain. Sampling the Hopfield model (chapter 6) differs from this chain in update order and interaction strength; chapter 7 shows the measured one-pattern ratio and what it does to the test.

The Curie–Weiss model has only two modes. The paper also tests the law on a Hopfield model that stores p random patterns; each pattern and its negative is a mode. You might expect competition to make the sampler worse at honouring the model’s preference.

The paper predicts the opposite. The field acts on pattern 1, so c is measured only on the kept runs: those that end in pattern 1 or its negative. If the largest early overlap decides, those runs won with a larger-than-usual overlap, and a larger overlap carries a stronger sign preference; c rises by an amplification factor A_p.

The paper approximates the early overlaps by independent Gaussians with a common variance, the field shifting only the mean of overlap 1, as in the Curie–Weiss model. It assumes the largest in size decides the pattern, and takes the Hopfield model’s own preference to be the Curie–Weiss one. Small p/N guarantees neither this equality nor independent overlaps (Sec. 4).

Approximate the p early overlaps by independent Gaussians with a common spread, and assume the largest in magnitude decides the pattern. Sign log-odds grow with the magnitude, so, to first order in the field, the kept runs express the preference more:

p	1	2	3	4
Ap	1	√2 = 1.414	1.662	1.836

Here c_CW is the Curie–Weiss value of c, computed numerically for the Hopfield protocol. The p = 2 cross-check is in the drawer below.

The paper tests the competition prediction on runs that choose a mode again, not on fresh starts (Sec. 4). For G > 0 the filtered start imitates a sampler that settled away from the favoured mode; reheating at β = 0.7, where the drift reduces an imbalance, must undo that choice, and the final cooling chooses again. The removal is applied at both field signs, so paired fields ±G follow one protocol.

For a symmetric start, flipping every spin and the sign of G would give q_chain(G) + q_chain(−G) = 1; the removal breaks this symmetry, and its memory could shift the sum. The measured sums are consistent with one but do not establish equivalence to fresh starts (App. B).

The steps: after two sweeps at each of the ten schedule temperatures, every run labelled (pattern 1, +) is removed, for both signs of the field. The rest reheat for 30 sweeps at β = 0.7, below the transition, with the patterns and field unchanged; then the final cooling follows the 834-sweep schedule, which begins at the hotter β = 0.6.

The Curie–Weiss reference c_CW runs through the same removal and reheating. For Curie–Weiss, 30 sweeps at β = 0.7 shrink the slowest deviation to about 1.2 × 10⁻⁴; this does not bound the memory of the filtered Hopfield runs (App. B).

Fig. M is an illustration, not this protocol: it follows the large-N drift with a field exaggerated to h = 0.05, so that you can see one well remain below β = 1.

Earlier chapters produced two numbers without the Hopfield runs: c_CW = 0.120, computed for the Curie–Weiss sampler through the same steps as the network runs (a brief cooling, removal of the runs labelled pattern 1 positive to leave the filtered start, reheating at β = 0.7, then the final cooling; Where the runs start), and A₃ = 1.662 for three patterns. Their product, 0.120 × 1.662 = 0.199, is the prediction. The figure sets it against the measurement, at the paper’s setting and at its neighbours, and lets you multiply by the one-pattern ratio to see what it changes.

At eight field strengths the sampler’s log-odds are plotted against the model’s; a faithful sampler would put every point on the diagonal. The slope of the line through them is c. At the strongest field, 66.9% of pattern-1 samples are positive against a model reference of 97.26%, so their logits give c = 0.70/3.57 ≈ 0.20; that reference is estimated from the mean overlap in the positive mode, so it is approximate (Sec. 4, App. B). The full figure is in the Plot drawer.

• The ratio 1.063 mixes update order with the 1/N against 1/(N−1) interaction strength. On that base the prediction is 0.212, outside the interval, so errors in the approximations may cancel (Sec. 5). Limits 2 gives the ratio at every N.
• With the ratio applied, N = 1,024 and 4,096 still miss by 18% and 4%; the size trend is not explained (Limits 3).
• The table’s p = 3 row (0.201) is a separate run at one field strength, both signs pooled, twelve pattern sets, with the Curie–Weiss reference; the headline 0.199 is the slope over eight fields and twenty pattern sets with per-set references.
• The 95% intervals resample pattern sets; they exclude the uncertainty of the model reference and a chosen 1% allowance (Fig. 2 caption; App. B).