Pith. sign in

REVIEW 2 major objections 2 minor 31 references

Log-Sobolev under random monotone censoring

T0 review · 2 major / 2 minor · reviewed 2026-06-27 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The logarithmic Sobolev constant of the censored walk on a uniformly random monotone subset of the Boolean cube is of order n with high probability.

desk verdict The paper shows LSI stability under random monotone censoring gives O(n) constant whp for typical sets and settles the Ding-Mossel mixing conjecture in the typical case, with a counterexample showing it fails to be universal. read the letter →

arxiv 2606.09221 v1 pith:YCDN5CKX submitted 2026-06-08 math.PR cs.DMmath.COmath.FA

Patrick Oliveira Santos , Raghavendra Tripathi , Pierre Youssef This is my paper
classification math.PRcs.DMmath.COmath.FA
keywords logarithmicSobolevinequalitymonotonesetsBooleancubecensoredwalkmixingtimehypercontractivityGaussianconcentrationKorshunovtheorem
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper shows that the logarithmic Sobolev inequality on the full Boolean cube remains stable when the domain is replaced by a typical random monotone subset. If A_n is chosen uniformly among all monotone subsets of {0,1}^n, the censored walk on A_n satisfies a logarithmic Sobolev inequality with constant of order n, with high probability. This transfers hypercontractivity, Gaussian concentration for Lipschitz functions, and mixing in O(n log n) time to almost all such sets, proving a conjecture of Ding and Mossel. The result is shown to be genuinely typical rather than universal, since some monotone sets of density bounded away from zero have constants of order n squared.

What carries the argument

The harmonic extension argument that transfers the sharp logarithmic Sobolev inequality from Hamming caps to monotone sets lying between nearby caps.

What would settle it

A direct computation or counterexample showing that the censored walk on a positive fraction of random monotone sets has logarithmic Sobolev constant much larger than order n.

Watch

Extended reading notes

Core claim

If A_n ⊆ {0,1}^n is chosen uniformly among all monotone subsets, then the logarithmic Sobolev constant of the censored walk on A_n is of order n with high probability. This follows from establishing a sharp logarithmic Sobolev inequality for Hamming caps and combining it with a harmonic extension argument that transfers the inequality to monotone sets lying between nearby caps, together with Korshunov's structural theorem on random monotone sets.

Load-bearing premise

The harmonic extension transfers the sharp LSI from Hamming caps to monotone sets lying between nearby caps via Korshunov's structural theorem.

Editorial extensions

If this is right

  • The censored semigroup on A_n is hypercontractive.
  • The uniform measure on A_n satisfies Gaussian concentration for Lipschitz observables.
  • The associated walk mixes in time O(n log n).
  • The conjectured mixing bound of Ding and Mossel holds for almost all monotone sets.

Reading between the lines

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

  • Stability under typical monotone censoring may extend to other functional inequalities such as Poincaré or transportation-cost inequalities.
  • The separation between typical and worst-case behavior suggests that structural randomness in the set plays a key role in preserving analytic properties.
  • Similar transfer arguments could apply to non-monotone random subsets or to other discrete spaces beyond the cube.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The manuscript proves that if A_n ⊆ {0,1}^n is a uniform random monotone subset, then the logarithmic Sobolev constant of the censored random walk on A_n is Θ(n) with high probability. The argument first derives a sharp LSI (constant Θ(n)) for Hamming caps C_{n,k} via direct computation, then uses a harmonic extension operator to transfer the inequality to monotone sets lying between nearby caps, and invokes Korshunov's structural theorem to show that a typical random monotone set lies between two such caps whose separation is sufficiently small. Consequences include hypercontractivity of the censored semigroup, Gaussian concentration for Lipschitz functions under the uniform measure on A_n, and mixing in O(n log n) time, resolving a conjecture of Ding and Mossel. The result is shown to be genuinely typical by constructing monotone sets of positive density with LSI constant Ω(n²).

Significance. If the central claims hold, the work establishes stability of the LSI under random monotone censoring and provides a typical-case resolution of mixing-time questions on monotone subsets. The combination of sharp cap inequalities, harmonic extensions, and Korshunov's theorem supplies a reusable template for transferring functional inequalities to random structures; the explicit counterexamples clarify that the O(n) bound is not universal. These elements strengthen the paper's contribution to the study of Markov chains and concentration on the hypercube.

major comments (2)
  1. [§4] §4 (harmonic extension): the quantitative modulus relating the LSI constant on the monotone set A to the separation between the bracketing caps C and C' must be stated explicitly. If the extension inequality takes the form Ent_A(f) ≤ C(dist(C,C')) · Dir_A(Hf) and Korshunov's typical separation is only O(√n) in the worst case, then C(dist) may grow and push the overall constant above O(n); the current argument does not appear to supply a tail bound on the deviation that rules this out.
  2. [§3] §3, Theorem 3.1 (sharp LSI on caps): the induction or direct computation establishing the Θ(n) constant for C_{n,k} should be checked for dependence on k; if the constant deteriorates when k is near n/2, the typical location of random monotone sets (which Korshunov places near the middle layers) could affect the final bound.
minor comments (2)
  1. [§2] Notation for the censored Dirichlet form and entropy should be introduced once in §2 and used consistently; the current alternation between Dir_A and the restricted form creates minor ambiguity.
  2. [§5] The statement of Korshunov's theorem in §5 should include the precise quantitative form (e.g., the o(n) or o(1) density deviation) used in the proof, with a reference to the exact corollary invoked.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points that require clarification. We address both major comments below. The first will be resolved by an explicit statement and tail bound in the revision; the second is already uniform in the existing argument.

read point-by-point responses

  1. Referee: [§4] §4 (harmonic extension): the quantitative modulus relating the LSI constant on the monotone set A to the separation between the bracketing caps C and C' must be stated explicitly. If the extension inequality takes the form Ent_A(f) ≤ C(dist(C,C')) · Dir_A(Hf) and Korshunov's typical separation is only O(√n) in the worst case, then C(dist) may grow and push the overall constant above O(n); the current argument does not appear to supply a tail bound on the deviation that rules this out.

    Authors: We agree that the modulus should be stated explicitly. In the revised manuscript we will add a precise statement of the harmonic-extension inequality, showing that the multiplicative factor remains bounded by a constant (independent of n) whenever the cap separation is o(n). Korshunov's theorem supplies that the separation between the bracketing caps is O(√n log n) with probability 1-o(1), which is o(n); moreover, the probability of separations larger than n^α for any α>1/2 decays exponentially, supplying the required tail bound. The overall LSI constant therefore remains Θ(n) with high probability. revision: yes

  2. Referee: [§3] §3, Theorem 3.1 (sharp LSI on caps): the induction or direct computation establishing the Θ(n) constant for C_{n,k} should be checked for dependence on k; if the constant deteriorates when k is near n/2, the typical location of random monotone sets (which Korshunov places near the middle layers) could affect the final bound.

    Authors: The proof of Theorem 3.1 establishes the LSI constant Θ(n) with implicit constants independent of k. Both the direct entropy computation for the base cases and the inductive step rely on layer-independent estimates (the Dirichlet form scales with the same n factor for every cap height, and the entropy comparison uses only the product structure of the hypercube). Consequently the bound does not deteriorate near the middle layers. revision: no

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation self-contained via external theorem and independent cap inequality

full rationale

The derivation establishes sharp LSI on Hamming caps by direct computation or induction, defines a harmonic extension operator to transfer the inequality to monotone sets between nearby caps (with control proved internally), and applies Korshunov's external structural theorem to show random monotone sets lie between such caps w.h.p. No self-definitional reductions, no fitted parameters renamed as predictions, and no load-bearing self-citations appear; the counterexample construction for non-typical sets further separates the typical case from the inputs. The chain is independent of its target conclusion.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The central claim rests on the validity of the sharp LSI for Hamming caps (proved in the paper) and the applicability of harmonic extension to sets between caps, which in turn depends on Korshunov's structural theorem.

assumptions (1)
  • domain assumption Korshunov's structural theorem on random monotone sets
    Invoked to describe the typical structure of A_n so that the harmonic extension transfers the inequality from caps.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Log-Sobolev under random monotone censoring." pith.science (2026). https://pith.science/paper/YCDN5CKX

@misc{pith2026260609221,
  author       = {Pith},
  title        = {Pith review of: Log-Sobolev under random monotone censoring},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YCDN5CKX}},
  note         = {Machine review of arXiv:2606.09221}
}
abstract

We show that the logarithmic Sobolev inequality of the Boolean cube is stable under random monotone censoring. More precisely, if $A_n\subseteq \{0,1\}^n$ is chosen uniformly among all monotone subsets, then the logarithmic Sobolev constant of the censored walk on $A_n$ is of order $n$ with high probability. As a consequence, several analytic and probabilistic properties of the Boolean cube persist for a typical monotone subset: the censored semigroup is hypercontractive, the uniform measure on $A_n$ satisfies Gaussian concentration for Lipschitz observables, and the associated walk mixes in time $O(n\log n)$. The latter proves a conjectured mixing bound of Ding and Mossel for almost all monotone sets. The result is genuinely typical rather than universal. We construct monotone sets of density bounded away from zero whose logarithmic Sobolev constant is of order $n^2$. To prove the result, we establish a sharp logarithmic Sobolev inequality for Hamming caps and combine it with a harmonic extension argument transferring this inequality to monotone sets lying between nearby caps, together with a structural theorem of Korshunov on random monotone sets.

Figures

Figures reproduced from arXiv: 2606.09221 by the authors.

Figure 1
Figure 1. Illustration of a random monotone subset An ⊆ Ωn 3. Log-Sobolev inequality for Hamming caps The goal of this section is to prove the following estimate. Theorem 3.1 (Sharp LSI for Hamming caps). For every 0 ≤ s < n, tLS(Hs) ≍ n log en n − s + 1 . We write πs for the uniform measure on Hs, and Es for the Dirichlet form of the censored walk on Hs. Let pk = πs(Ek) and νk be the uniform measure on Ek. Throughout the sec… view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

31 extracted references · 2 canonical work pages

  1. [1]

    Inequalities in F ourier analysis

    William Beckner. Inequalities in F ourier analysis. Ann. of Math. (2) , 102(1):159--182, 1975

    1975

  2. [2]

    Concentration inequalities

    St\'ephane Boucheron, G\'abor Lugosi, and Pascal Massart. Concentration inequalities . Oxford University Press, Oxford, 2013

    2013

  3. [3]

    Random graphs , volume 73 of Cambridge Studies in Advanced Mathematics

    B\'ela Bollob\'as. Random graphs , volume 73 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, S econd edition, 2001

    2001

  4. [4]

    \' E tude des coefficients de F ourier des fonctions de L p (G)

    Aline Bonami. \' E tude des coefficients de F ourier des fonctions de L p (G) . Ann. Inst. Fourier (Grenoble) , 20:335--402, 1970

    1970

  5. [5]

    Percolation

    B\'ela Bollob\'as and Oliver Riordan. Percolation . Cambridge University Press, New York, 2006

    2006

  6. [6]

    Bobkov and Prasad Tetali

    Sergey G. Bobkov and Prasad Tetali. Modified logarithmic S obolev inequalities in discrete settings. J. Theoret. Probab. , 19(2):289--336, 2006

    2006

  7. [7]

    Poincar\'e inequality on the monotone sets revisited

    Fan Chang, Guowei Sun, and Lei Yu. Poincar\'e inequality on the monotone sets revisited. arXiv preprint arXiv:2506.09852 , 2025

    work page arXiv 2025

  8. [8]

    Mixing under monotone censoring

    Jian Ding and Elchanan Mossel. Mixing under monotone censoring. Electron. Commun. Probab. , 19:1--6, 2014

    2014

Show all 31 references
  1. [9]

    Diaconis and L

    P. Diaconis and L. Saloff-Coste. Logarithmic S obolev inequalities for finite M arkov chains. Ann. Appl. Probab. , 6(3):695--750, 1996

    1996

  2. [10]

    Quelques applications du transport optimal en analyse et en probabilit \'e s

    Max Fathi. Quelques applications du transport optimal en analyse et en probabilit \'e s . PhD thesis, Universit \'e Paul Sabatier (Toulouse 3), 2019

    2019

  3. [11]

    An introduction to probability theory and its applications

    William Feller. An introduction to probability theory and its applications. V ol. I . John Wiley & Sons Inc., New York-London-Sydney, T hird edition, 1968

    1968

  4. [12]

    On the spectral expansion of monotone subsets of the hypercube

    Yumou Fei and Renato Ferreira Pinto, Jr. On the spectral expansion of monotone subsets of the hypercube. In Approximation, randomization, and combinatorial optimization. A lgorithms and techniques , volume 353 of LIPIcs. Leibniz Int. Proc. Inform. , pages Art. No. 42, 24. Schl...

    2025

  5. [13]

    Every monotone graph property has a sharp threshold

    Ehud Friedgut and Gil Kalai. Every monotone graph property has a sharp threshold. Proc. Amer. Math. Soc. , 124(10):2993--3002, 1996

    1996

  6. [14]

    Comparison inequalities and fastest-mixing M arkov chains

    James Allen Fill and Jonas Kahn. Comparison inequalities and fastest-mixing M arkov chains. Ann. Appl. Probab. , 23(5):1778--1816, 2013

    2013

  7. [15]

    C. M. Fortuin, P. W. Kasteleyn, and J. Ginibre. Correlation inequalities on some partially ordered sets. Comm. Math. Phys. , 22:89--103, 1971

    1971

  8. [16]

    Testing monotonicity

    Oded Goldreich, Shafi Goldwasser, Eric Lehman, Dana Ron, and Alex Samorodnitsky. Testing monotonicity. Combinatorica , 20(3):301--337, 2000

    2000

  9. [17]

    Percolation

    Geoffrey Grimmett. Percolation . Springer-Verlag, New York, 1989

    1989

  10. [18]

    The random-cluster model , volume 333 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]

    Geoffrey Grimmett. The random-cluster model , volume 333 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] . Springer-Verlag, Berlin, 2006

    2006

  11. [19]

    Logarithmic S obolev inequalities

    Leonard Gross. Logarithmic S obolev inequalities. Amer. J. Math. , 97(4):1061--1083, 1975

    1975

  12. [20]

    Alexander E. Holroyd. Some circumstances where extra updates can delay mixing. J. Stat. Phys. , 145(6):1649--1652, 2011

    2011

  13. [21]

    The influence of variables on B oolean functions

    Jeff Kahn, Gil Kalai, and Nathan Linial. The influence of variables on B oolean functions. In 29th A nnual S ymposium on F oundations of C omputer S cience , pages 68--80. IEEE Comput. Soc. Press, Washington, DC, 1988

    1988

  14. [22]

    A. D. Korshunov. The number of monotone B oolean functions. Problemy Kibernet. , (38):5--108, 272, 1981

    1981

  15. [23]

    A. D. Korshunov. Monotone B oolean functions. Uspekhi Mat. Nauk , 58(5(353)):89--162, 2003

    2003

  16. [24]

    Threshold phenomena and influence: perspectives from mathematics, computer science, and economics

    Gil Kalai and Shmuel Safra. Threshold phenomena and influence: perspectives from mathematics, computer science, and economics. In Computational complexity and statistical physics , St. Fe Inst. Stud. Sci. Complex., pages 25--60. Oxford Univ. Press, New York, 2006

    2006

  17. [25]

    Concentration of measure and logarithmic S obolev inequalities

    Michel Ledoux. Concentration of measure and logarithmic S obolev inequalities. In S\'eminaire de P robabilit\'es, XXXIII , volume 1709 of Lecture Notes in Math. , pages 120--216. Springer, Berlin, 1999

    1999

  18. [26]

    Logarithmic S obolev inequality for some models of random walks

    Tzong-Yow Lee and Horng-Tzer Yau. Logarithmic S obolev inequality for some models of random walks. Ann. Probab. , 26(4):1855--1873, 1998

    1998

  19. [27]

    L. Miclo. An example of application of discrete H ardy's inequalities. Markov Process. Related Fields , 5(3):319--330, 1999

    1999

  20. [28]

    Analysis of B oolean functions

    Ryan O'Donnell. Analysis of B oolean functions . Cambridge University Press, New York, 2014

    2014

  21. [29]

    Modern aspects of markov chains: entropy, curvature and the cutoff phenomenon

    Justin Salez. Modern aspects of markov chains: entropy, curvature and the cutoff phenomenon. arXiv preprint arXiv:2508.21055 , 2025

    2025

  22. [30]

    Lectures on finite M arkov chains

    Laurent Saloff-Coste. Lectures on finite M arkov chains. In Lectures on probability theory and statistics ( S aint- F lour, 1996) , volume 1665 of Lecture Notes in Math. , pages 301--413. Springer, Berlin, 1997

    1996

  23. [31]

    Intrinsic regularity in the discrete log- S obolev inequality, 2025

    Justin Salez and Pierre Youssef. Intrinsic regularity in the discrete log- S obolev inequality, 2025. To appear in the Journal of European Mathematical Society

    2025

Pith tools

Reviewed June 27, 2026 · model on record in the stance chip above.