David Ellis, Yuval Filmus and Ehud Friedgut

Random Structures and Algorithms, Volume 46, Issue 3, 2015, pp. 494–530

We prove that a balanced Boolean function on $S_n$ whose Fourier transform is highly concentrated on the first two irreducible representations of $S_n$ 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 $S_n$ 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, the preceding paper deals with Boolean functions of expectation $O(1/n)$ whose Fourier transform is highly concentrated on the first two irreducible representations of $S_n$. These need not be close to dictatorships; rather, they must be close to a union of a constant number of cosets of point-stabilizers.

@article{EFF2,

author = {David Ellis and Yuval Filmus and Ehud Friedgut},

title = {A stability result for balanced dictatorships in {$S_n$}},

journal = {Random Structures and Algorithms},

volume = {46},

number = {3},

year = {2015},

pages = {494--530}

}

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author = {David Ellis and Yuval Filmus and Ehud Friedgut},

title = {A stability result for balanced dictatorships in {$S_n$}},

journal = {Random Structures and Algorithms},

volume = {46},

number = {3},

year = {2015},

pages = {494--530}

}