Random Energy Model with Compact Distributions

Nabin Kumar Jana

REVIEW

Random Energy Model with Compact Distributions

Nabin Kumar Jana

Probability · Submitted 28 Feb 2006 · 6 references

Pith's verdict: A correct and genuinely new limit theorem for compact-support REM free energy; the polynomial-tail condition does the work, and while two typos need fixing, the paper deserves peer review.

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arXiv IDmath/0602666 (version 1)

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CitationNabin Kumar Jana. "Random Energy Model with Compact Distributions." arXiv:math/0602666 (2006). https://pith.science/paper/3UNJOVNC

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Editor's note

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The paper does something real: it extends the REM free-energy asymptotics from the Gaussian case to any compactly supported symmetric distribution whose tail at the boundary of the support decays at most polynomially. Theorem 2.1 is stated cleanly, the hypothesis on a_N(s) is explicit, and the proof is essentially complete. The upper bound via Jensen is immediate; the lower bound follows Talagrand's second-moment argument, and it works because the polynomial-tail condition guarantees both 2^N a_N(s) -> infinity and log a_N(s)/T_N -> 0. The examples are a nice bonus: uniform, truncated double exponential, Bernoulli, and a few others. This is a solid extension of the solvable class, not a new technique, and the paper does not oversell it. The soft spots are minor in proportion. Equation (7) prints log Z_N > max(-H_N) where the argument needs log Z_N >= beta max(-H_N) >= beta s T_N. The missing beta is clearly a typo, and the positivity estimate still works once inserted. Example 3.6's density is not normalized as printed; it should be N/[2(e^{N T_N}-1)] e^{N|x|} on [-T_N,T_N]. Again a typo, since the claimed a_N(s) -> 1/2 is right with the corrected constant. The theorem statement says the variables have a density, but Example 3.3 is a Bernoulli (atomic) distribution; the proof only uses symmetry and probabilities, so this is a harmless inconsistency, though the statement should either say 'distribution' or allow atoms. Remark 3.1, asking about almost-sure convergence, is left open and that is honest. I have no load-bearing objection. The alpha = infinity case is handled a bit tersely, but the lower-bound argument gives divergence and the upper bound is trivial, so the claim is fine. The polynomial-tail condition is restrictive, but that is precisely what distinguishes this regime from the Gaussian REM, where faster decay produces a phase transition; so it is not an artificial assumption. This paper is for mathematical physicists and probabilists working on spin-glass models and large deviations. It deserves a serious referee: the core theorem is correct, the typos are fixable, and the extension is worth publishing. I would send it out and expect a minor-revision outcome.

Referee report

Minor objections (6)

[Eq. (7), p. 3] In Eq. (7), the printed inequality log Z_N(β) > max_σ{-H_N(σ)} is false in general; the valid bound is log Z_N(β) ≥ β max_σ{-H_N(σ)} ≥ βsT_N > 0, and the missing factor of β should be inserted.
[Example 3.6, p. 5] The displayed density in Example 3.6 is not normalized; the normalization constant should be N/[2(e^{N T_N}-1)] rather than N/[2(e^{T_N}-1)] to make the integral over [-T_N,T_N] equal to 1, after which the stated convergence a_N(s)→1/2 follows.
[Example 3.3, p. 4] Example 3.3 uses Bernoulli point masses at ±T_N, whereas Theorem 2.1 is stated for random variables with a density φ_N; since the proof only uses symmetry and the tail probabilities, the example is valid under an immediate extension, but the text should reconcile this with the theorem's hypothesis.
[Example 3.5, p. 5] In Example 3.5, the density is supported on [-πT_N/2, πT_N/2], so T_N is not the support radius used in Theorem 2.1; either relabel the scale parameter or adjust the support interval to match the notation of the theorem.
[Proof of Theorem 2.1, α=∞ case, p. 3] The α=∞ case is compressed into the sentence 'the above argument shows that lim inf ... = ∞'; the conclusion is correct because (N/T_N)log2 dominates the possible -m log N/T_N term, but a short justification would improve readability.
[Remark 3.1, p. 5] Remark 3.1 explicitly leaves open the almost-sure analogue of the theorem; this is a clearly stated limitation and does not affect the expectation result claimed in Theorem 2.1.

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The paper

The paper claims

Compactly supported random energy models have an explicit free-energy limit, $\alpha \log 2 + \beta$, under a polynomial-tail condition.

The reading

The paper proves that for the Random Energy Model with compactly supported, symmetric Hamiltonians, the expected free energy has an explicit asymptotic whenever the upper-tail probabilities of the Hamiltonian decay no faster than polynomially. If $N/T_N \to \alpha$, then $(1/T_N) \mathbb{E} \log Z_N(\beta) \to \alpha \log 2 + \beta$ for each $\beta > 0$; when $\alpha$ is finite, this is the same as $(1/N) \mathbb{E} \log Z_N(\beta) \to \log 2 + \beta/\alpha$. The result covers uniform, truncated double-exponential, two-point, and polynomial-triangle densities, and it shows that the limiting free energy is linear in $\beta$, so there is no phase transition under these hypotheses. The interest is that the REM is the standard toy model for spin glasses, and the Gaussian case was solvable while the compact non-Gaussian case had no general evaluation.

What carries the argument

The central objects are the tail probabilities $a_N(s) = \mathbb{P}\{H_N \geq s T_N\}$ and the scale ratio $N/T_N$. The upper bound is Jensen's inequality: $\mathbb{E} \log Z_N \leq \log \mathbb{E} Z_N \leq N \log 2 + \beta T_N$, so dividing by $T_N$ gives $\alpha \log 2 + \beta$ in the limit. The lower bound counts the set $X_N$ of configurations with $-H_N(\sigma) \geq sT_N$; the second-moment estimate $\mathbb{E}(X_N - \mathbb{E}X_N)^2 \leq 2^N a_N(s)$ shows $X_N$ is usually close to $2^N a_N(s)$, and the polynomial-tail condition forces both $2^N a_N(s) \to \infty$ and $\log a_N(s)/T_N \to 0$. Taking $s \to 1$ then closes the gap.

Extended reading notes

Core claim

The central claim is Theorem 2.1. For each $N$, take $2^N$ independent symmetric random variables $H_N(\sigma)$ with a density supported in $[-T_N, T_N]$, and set $a_N(s) = \mathbb{P}\{H_N \geq s T_N\}$. If for every $0 < s < 1$ there is an $m \geq 0$ such that $0 < \lim_{N\to\infty} N^m a_N(s) < \infty$, and $N/T_N \to \alpha \in [0,\infty]$, then for every $\beta > 0$, $\lim_{N\to\infty} (1/T_N) \mathbb{E} \log Z_N(\beta) = \alpha \log 2 + \beta$. Equivalently, when $\alpha$ is finite, the free energy per configuration is $\log 2 + \beta/\alpha$. The proof does not require Gaussian or exponential-tail estimates; it uses only symmetry, compact support, and the polynomial-tail condition, and it applies to a list of concrete densities.

Load-bearing premise

The whole result depends on the upper-tail probability $\mathbb{P}\{H_N \geq sT_N\}$ decaying no faster than a polynomial in $N$; if the tail is thinner, the lower-bound count of high-energy configurations can vanish and the stated formula need not hold.

Editorial extensions

If this is right

For uniform $H_N$ on $[-N,N]$, the theorem gives $\lim_{N\to\infty} \frac{1}{N} \mathbb{E} \log Z_N(\beta) = \log 2 + \beta$.
For support widths $T_N = N^\gamma$ with $\gamma > 1$ or $T_N = 2^N$, the normalized free energy per $T_N$ tends to $\beta$: the entropy term $\alpha \log 2$ vanishes because $\alpha = 0$.
For support widths $T_N = \log N$ or $T_N = N^\gamma$ with $\gamma < 1$, the normalized free energy per $T_N$ diverges, reflecting the overwhelming influence of the growing support.
The formula is linear in $\beta$ for every $\beta > 0$ under the hypotheses, so the model shows no phase transition in this regime.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same pair of estimates should give quenched convergence of $\frac{1}{T_N} \log Z_N(\beta)$ in probability, not just in expectation, provided a concentration argument for $\log Z_N$ can be supplied; the paper leaves this open.
Because the formula is independent of the particular shape of the density once the tail is polynomial, it suggests a universality class: any compact symmetric density whose upper tail is regularly varying with any exponent yields the same limit, with the exponent affecting only finite-$N$ corrections.
The lower-bound counting method may extend to Generalized Random Energy Models with compactly supported energies, where $\log 2$ would be replaced by the entropy of the branching tree.

Original abstract

In this paper we study the Random energy model - so called toy model of the spin glass theory - where the underlying distributions are compactly supported. We prove a general theorem on the asymptotics of free energy and obtain formulae in several interesting cases - like uniform distribution, truncated double exponential.

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In this paper we study the Random energy model - so called toy model of the spin glass theory - where the underlying distributions are compactly supported. We prove a general theorem on the asymptotics of free energy and obtain formulae in several interesting cases - like uniform distribution, truncated double exponential.

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s0	In this paper we study the Random energy model - so called toy model of the spin glass theory - where the underlying distributions are compactly supported.	contribution 0.63	0.58	0.05	asserted 0.90
s1	We prove a general theorem on the asymptotics of free energy and obtain formulae in several interesting cases - like uniform distribution, truncated double exponential.	contribution 0.52	0.96	0.80	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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References

Works cited (6)

[1]

Bovier, A. (2001). Statistical Mechanics of Disordered Systems, MaPhySto Lec- ture Notes , 192pp

View on Pith 2001
[2]

Contucci, P., Esposti, M. D., Giardin `a, C. and Graffi, S. (2003). Thermody- namical Limit for Corelated Gaussian Random Energy Models. Commun. Math. Phys., 236, 55–63

View on Pith 2003
[3]

Derrida, B. (1981). Random Energy Model: An Exactly Solvable Model of Di s- ordered Systems. Phys. Rev. , B24, 2613–2626

View on Pith 1981
[4]

Dorlas, T. C. and Wedagedera, J. R. (2001). Large Deviations and The Ran- dom Energy Model. Int. J. of Mod. Phy. B , 15, No. 1, 1–15. Random Energy Model 6

View on Pith 2001
[5]

Guerra, F. and Toninelli, F. L. (2002). The Thermodynamic Limit in Mean Field Spin Glass Models. Commun. Math. Phys. , 230, 71–79

View on Pith 2002
[6]

Talagrand, M. (2003). SPIN GLASSES: A Challenge for Mathematicians , Spinger-Verlag, New York

View on Pith 2003

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]

Bovier, A. (2001). Statistical Mechanics of Disordered Systems, MaPhySto Lec- ture Notes , 192pp

View on Pith 2001

[2] [2]

Contucci, P., Esposti, M. D., Giardin `a, C. and Graffi, S. (2003). Thermody- namical Limit for Corelated Gaussian Random Energy Models. Commun. Math. Phys., 236, 55–63

View on Pith 2003

[3] [3]

Derrida, B. (1981). Random Energy Model: An Exactly Solvable Model of Di s- ordered Systems. Phys. Rev. , B24, 2613–2626

View on Pith 1981

[4] [4]

Dorlas, T. C. and Wedagedera, J. R. (2001). Large Deviations and The Ran- dom Energy Model. Int. J. of Mod. Phy. B , 15, No. 1, 1–15. Random Energy Model 6

View on Pith 2001

[5] [5]

Guerra, F. and Toninelli, F. L. (2002). The Thermodynamic Limit in Mean Field Spin Glass Models. Commun. Math. Phys. , 230, 71–79

View on Pith 2002

[6] [6]

Talagrand, M. (2003). SPIN GLASSES: A Challenge for Mathematicians , Spinger-Verlag, New York

View on Pith 2003