Harmonicity and invariance on slices of the Boolean cube

Filmus, Yuval; Mossel, Elchanan

doi:10.1007/s00440-019-00900-w

Cited by 27 publications

(32 citation statements)

References 34 publications

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“…It turns out that every random variable of the form f (ξ), for ξ ∈ BL(n, k), can be represented in the form g(ξ), for g a harmonic multilinear polynomial. We will moreover need the fact that if f is a polynomial of degree d, then g also has degree at most d. The following lemma effectively appears as [17,Lemma 3.17].…”

Section: "Weak" Anticoncentration Via Hypercontractivitymentioning

confidence: 99%

“…We will moreover need the fact that if

f

is a polynomial of degree

d

, then

g

also has degree at most

d

. The following lemma effectively appears as [, Lemma 3.17]. Lemma Let

f

be an

n

‐variable polynomial of degree

d

, and let

bold-italicξ \in BL (n, k)

.…”

Section: Probabilistic Toolsmentioning

confidence: 99%

“…Many standard probabilistic tools are designed to work for product distributions, where independence can be exploited. Although

BL (n, k)

is not a product distribution, there is a well‐known way to approximate

BL (n, k)

with a product of Bernoulli‐distributed random variables, and for many purposes these distributions can be considered essentially equivalent (see for example [, Corollary 1.16] and the invariance principles in ). However, for the purposes of proving Theorems and , this kind of approximation is too coarse.…”

Section: Probabilistic Toolsmentioning

confidence: 99%

“…It would be interesting even to prove a bound of the form

{prefixind}_{r} false(k, ℓ false) ⩽ 1 - ε + o_{k} false(1 false)

for any

ε > 0

that does not depend on

r

; perhaps this can be accomplished with a careful bare hands fourth‐moment computation as in . We remark however that the bound

{prefixind}_{r} false(k, ℓ false) ⩽ 1 - 2^{- 4 / 3} 3^{- 16 r}

in Theorem is clearly not best possible; instead of using hypercontractivity for

BL (n, n / 2)

via Lemma , it would have been possible to give a less direct proof using the much more developed theory of hypercontractivity for

{Rad}^{n}

, then appealing to the invariance principle in . This would have given the bound

{prefixind}_{r} false(k, ℓ false) ⩽ 1 - 9^{- r} + o_{k} false(1 false)

.…”

Section: Further Directions Of Researchmentioning

confidence: 99%

“…There is also the possibility that more natural anticoncentration theorems could be stated in terms of harmonic polynomials (recall the definition from Section 2.2), which are in some sense the correct representation of functions on the slice (see for example ). For harmonic polynomials of

BL (n, p n)

random vectors, the invariance principle in can be used to apply standard anticoncentration theorems, but the error terms in this invariance principle prevent one from obtaining optimal bounds in this way.…”

Section: Further Directions Of Researchmentioning

confidence: 99%

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Anticoncentration for subgraph statistics

Kwan

Sudakov

Tran

2018

J. London Math. Soc.

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Consider integers k,ℓ such that 0⩽ℓ⩽0ptk2. Given a large graph G, what is the fraction of k‐vertex subsets of G which span exactly ℓ edges? When G is empty or complete, and ℓ is zero or ()k2, this fraction can be exactly 1. On the other hand, if ℓ is far from these extreme values, one might expect that this fraction is substantially smaller than 1. This was recently proved by Alon, Hefetz, Krivelevich, and Tyomkyn who initiated the systematic study of this question and proposed several natural conjectures. Let ℓ∗=minfalse{ℓ,()k2−ℓfalse}. Our main result is that for any k and ℓ, the fraction of k‐vertex subsets that span ℓ edges is at most prefixlogO(1)false(ℓ∗/kfalse)k/ℓ∗, which is best‐possible up to the logarithmic factor. This improves on multiple results of Alon, Hefetz, Krivelevich, and Tyomkyn, and resolves one of their conjectures. In addition, we also make some first steps towards some analogous questions for hypergraphs. Our proofs involve some Ramsey‐type arguments, and a number of different probabilistic tools, such as polynomial anticoncentration inequalities, hypercontractivity, and a coupling trick for random variables defined on a ‘slice’ of the Boolean hypercube.

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Section: "Weak" Anticoncentration Via Hypercontractivitymentioning

confidence: 99%

“…We will moreover need the fact that if

f

is a polynomial of degree

d

, then

g

also has degree at most

d

. The following lemma effectively appears as [, Lemma 3.17]. Lemma Let

f

be an

n

‐variable polynomial of degree

d

, and let

bold-italicξ \in BL (n, k)

.…”

Section: Probabilistic Toolsmentioning

confidence: 99%

“…Many standard probabilistic tools are designed to work for product distributions, where independence can be exploited. Although

BL (n, k)

is not a product distribution, there is a well‐known way to approximate

BL (n, k)

Section: Probabilistic Toolsmentioning

confidence: 99%

“…It would be interesting even to prove a bound of the form

{prefixind}_{r} false(k, ℓ false) ⩽ 1 - ε + o_{k} false(1 false)

for any

ε > 0

that does not depend on

r

; perhaps this can be accomplished with a careful bare hands fourth‐moment computation as in . We remark however that the bound

{prefixind}_{r} false(k, ℓ false) ⩽ 1 - 2^{- 4 / 3} 3^{- 16 r}

in Theorem is clearly not best possible; instead of using hypercontractivity for

BL (n, n / 2)

via Lemma , it would have been possible to give a less direct proof using the much more developed theory of hypercontractivity for

{Rad}^{n}

, then appealing to the invariance principle in . This would have given the bound

{prefixind}_{r} false(k, ℓ false) ⩽ 1 - 9^{- r} + o_{k} false(1 false)

.…”

Section: Further Directions Of Researchmentioning

confidence: 99%

BL (n, p n)

Section: Further Directions Of Researchmentioning

confidence: 99%

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Anticoncentration for subgraph statistics

Kwan

Sudakov

Tran

2018

J. London Math. Soc.

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Anticoncentration and Berry–Esseen bounds for random tensors

Dodos

Tyros

2023

Probab. Theory Relat. Fields

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Boolean Function Analysis on High-Dimensional Expanders

Dikstein,

Dinur,

Filmus

et al. 2024

Combinatorica

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We initiate the study of Boolean function analysis on high-dimensional expanders. We give a random-walk based definition of high-dimensional expansion, which coincides with the earlier definition in terms of two-sided link expanders. Using this definition, we describe an analog of the Fourier expansion and the Fourier levels of the Boolean hypercube for simplicial complexes. Our analog is a decomposition into approximate eigenspaces of random walks associated with the simplicial complexes. Our random-walk definition and the decomposition have the additional advantage that they extend to the more general setting of posets, encompassing both high-dimensional expanders and the Grassmann poset, which appears in recent work on the unique games conjecture. We then use this decomposition to extend the Friedgut–Kalai–Naor theorem to high-dimensional expanders. Our results demonstrate that a constant-degree high-dimensional expander can sometimes serve as a sparse model for the Boolean slice or hypercube, and quite possibly additional results from Boolean function analysis can be carried over to this sparse model. Therefore, this model can be viewed as a derandomization of the Boolean slice, containing only $$|X(k-1)|=O(n)$$ | X ( k - 1 ) | = O ( n ) points in contrast to $$\left( {\begin{array}{c}n\\ k\end{array}}\right) $$ n k points in the (k)-slice (which consists of all n-bit strings with exactly k ones).

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Harmonicity and invariance on slices of the Boolean cube

Abstract: In a recent work with Kindler and Wimmer we proved an invariance principle for the slice for lowinfluence, low-degree harmonic multilinear polynomials (a polynomial in x1, . . . , xn is harmonic if it is annihilated by

Cited by 27 publications

References 34 publications

Anticoncentration for subgraph statistics

Anticoncentration for subgraph statistics

Anticoncentration and Berry–Esseen bounds for random tensors

Boolean Function Analysis on High-Dimensional Expanders

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