Invariance Principle on the Slice

Probab. Theory Relat. Fields

Mossel

2019

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Section: Low-degree Polynomialsmentioning

confidence: 95%

“…It is natural to speculate that low-degree harmonic functions have similar distributions under ν and µ. Unfortunately, the proof of the invariance principle in [18] goes through Gaussian space, rendering the low-influence condition necessary even when comparing ν and µ.…”

Section: Low-degree Polynomialsmentioning

confidence: 99%

Harmonicity and invariance on slices of the Boolean cube

Probab. Theory Relat. Fields

Mossel

2019

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“…We bootstrap our approach by proving that every Boolean degree d function on [2n] n is a junta. The proof is a simple application of hypercontractivity, and already appears in [5]. We reproduce a simplified version here in order to make the paper self-contained.…”

Section: Bootstrappingmentioning

confidence: 99%

“…(We explain in Section 2 what degree d means for functions on the hypercube and on the slice. ) Filmus et al [5] proved a version of Theorem 1.1 (with a non-optimal bound on the number of points) when k/n is bounded away from 0, 1, but their bound deteriorates as k/n gets closer to 0, 1. We use their result (which we reproduce here, to keep the proof self-contained) to bootstrap our own inductive argument.…”

Section: Introductionmentioning

confidence: 99%

Boolean constant degree functions on the slice are juntas

Ihringer

2019

Discrete Mathematics

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“…. , n}; consult [25,26,29,30,39,51] for some of the work in this area. Recently, the Grassmann graph J q (n, k), which is the q-analog of the Johnson graph, has come to attention in theoretical computer science [15,16,40,41], but its research from the point of view of analysis of Boolean functions is at its infancy.…”

Section: Introductionmentioning

confidence: 99%

Boolean degree 1 functions on some classical association schemes

Journal of Combinatorial Theory, Series A

Ihringer

2019

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We investigate Boolean degree 1 functions for several classical association schemes, including Johnson graphs, Grassmann graphs, graphs from polar spaces, and bilinear forms graphs, as well as some other domains such as multislices (Young subgroups of the symmetric group). In some settings, Boolean degree 1 functions are also known as completely regular strength 0 codes of covering radius 1, Cameron-Liebler line classes, and tight sets.We classify all Boolean degree 1 functions on the multislice. On the Grassmann scheme Jq(n, k) we show that all Boolean degree 1 functions are trivial for n ≥ 5, k, n − k ≥ 2 and q ∈ {2, 3, 4, 5}, and that, for general q, the problem can be reduced to classifying all Boolean degree 1 functions on Jq(n, 2). We also consider polar spaces and the bilinear forms graphs, giving evidence that all Boolean degree 1 functions are trivial for appropriate choices of the parameters.1 (also strength 0 designs) [44]. Motivated by problems on permutation groups and finite geometry, Boolean degree 1 functions are also known as tight sets [2,12] and Cameron-Liebler line classes [17]. The history of Cameron-Liebler line classes is particularly complicated, as the problem was introduced by Cameron and Liebler [7], the term coined by Penttila [52,53], and the algebraic point of view as Boolean degree 1 functions only emerged later; see [59], in particular §3.3.1, for a discussion of this.Due to this variation of terminology, classification results of Boolean degree 1 functions on J(n, k) were obtained repeatedly in the literature (with small variations due to different definitions) at least three times, in [49] for completely regular strength 0 codes of covering radius 1, in [25] for Boolean degree 1 functions, and in [11] for Cameron-Liebler line classes:Theorem 1.2 (Folklore). Suppose that k, n − k ≥ 2. Every Boolean degree 1 function on J(n, k) is either constant or depends on a single coordinate.Depending on the definition used, this is either easy to observe or requires a more elaborate proof. For the hypercube H(n, 2), which is a product domain, and the Johnson graph J(n, k) classifying Boolean degree 1 functions is trivial, but there are various other classical association schemes for which classification is more difficult. The Grassmann scheme J q (n, k) consists of all k-spaces of F n q as vertices, two vertices being adjacent if their meet is a subspace of dimension k − 1. Boolean degree 1 functions on J q (4, 2) were intensively investigated, and many non-trivial examples [6,8,9,24,33,37] and existence conditions [34,47] are known.We call 1-dimensional subspaces of F n q points, 2-dimensional subspaces of F n q lines, and (n − 1)dimensional subspaces of F n q hyperplanes. For a point p we define p + (S) = 1 p∈S and p − (S) = 1 p / ∈S , and for a hyperplane π we define π + (S) = 1 S⊆π and π − (S) = 1 S π . The following was shown by Drudge for q = 3 [17, Theorem 6.4]; by Gavrilyuk and Mogilnykh for q = 4 [35, Theorem 3]; and by Gavrilyuk and Matkin [32,45] for q = 5; the result for q = 2 fol...