A quasi-stability result for dictatorships in S n

Ellis, David; Filmus, Yuval; Friedgut, Ehud

doi:10.1007/s00493-014-3027-1

Cited by 18 publications

(44 citation statements)

References 34 publications

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“…In contrast, [7] deals with Boolean functions of expectation O(1/n) whose Fourier transform is highly concentrated on the first two irreducible representations of Sn. These need not be close to dictatorships; rather, they must be close to a union of a constant number of cosets of point-stabilizers.…”

Section: Introductionmentioning

confidence: 99%

“…This paper (together with [7] and [8]) is part of a trilogy dealing with stability and 'quasi-stability' results concerning Boolean functions on the symmetric group, which are of 'low complexity', in a Fouriertheoretic sense.…”

Section: Introductionmentioning

confidence: 99%

“…The goal of the current paper, together with [7], is to provide stability versions of Theorem 1. This is in the spirit of similar projects in the Abelian case, which have proved extremely useful and applicable, see e.g.…”

Section: Introductionmentioning

confidence: 99%

“…A good example of this is Theorem 6 in this paper, where we characterize the almost-extremal sets for the edge-isoperimetric inequality in the transposition graph on S n (the Cayley graph on S n generated by the transpositions). See [7] for more about applications in the symmetric group setting.…”

Section: Introductionmentioning

confidence: 99%

“…The division between the three papers in our trilogy is as follows: in [7], we deal with Boolean functions which are close to U 1 , and have expectation O(1/n). We prove that such a functions must be close to a sum of dictatorships -equivalently, close to the characteristic function of a union of 1-cosets.…”

Section: Introductionmentioning

confidence: 99%

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A stability result for balanced dictatorships in S_n

Ellis

Filmus

Friedgut

2013

Random Struct. Alg.

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We prove that a balanced Boolean function on Sn whose Fourier transform is highly concentrated on the first two irreducible representations of Sn, is close in structure to a dictatorship, a function which is determined by the image or pre-image of a single element. As a corollary, we obtain a stability result concerning extremal isoperimetric sets in the Cayley graph on Sn generated by the transpositions.Our proof works in the case where the expectation of the function is bounded away from 0 and 1. In contrast, [7] deals with Boolean functions of expectation O(1/n) whose Fourier transform is highly concentrated on the first two irreducible representations of Sn. These need not be close to dictatorships; rather, they must be close to a union of a constant number of cosets of point-stabilizers.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A stability result for balanced dictatorships in S_n

Ellis

Filmus

Friedgut

2013

Random Struct. Alg.

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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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