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Perhaps the most important structural result about general large dense graphs is the Szemerédi regularity lemma. Here is a standard formulation of that lemma:
Lemma 1 (Szemerédi regularity lemma) Let be a graph on vertices, and let . Then there exists a partition for some with the property that for all but at most of the pairs , the pair is -regular in the sense that
whenever are such that and , and is the edge density between and . Furthermore, the partition is equitable in the sense that for all .
There are many proofs of this lemma, which is actually not that difficult to establish; see for instance these previous blog posts for some examples. In this post I would like to record one further proof, based on the spectral decomposition of the adjacency matrix of , which is essentially due to Frieze and Kannan. (Strictly speaking, Frieze and Kannan used a variant of this argument to establish a weaker form of the regularity lemma, but it is not difficult to modify the Frieze-Kannan argument to obtain the usual form of the regularity lemma instead. Some closely related spectral regularity lemmas were also developed by Szegedy.) I found recently (while speaking at the Abel conference in honour of this year’s laureate, Endre Szemerédi) that this particular argument is not as widely known among graph theory experts as I had thought, so I thought I would record it here.
For reasons of exposition, it is convenient to first establish a slightly weaker form of the lemma, in which one drops the hypothesis of equitability (but then has to weight the cells by their magnitude when counting bad pairs):
Lemma 2 (Szemerédi regularity lemma, weakened variant) . Let be a graph on vertices, and let . Then there exists a partition for some with the property that for all pairs outside of an exceptional set , one has
for some orthonormal basis of and some eigenvalues , which we arrange in decreasing order of magnitude:
We can compute the trace of as
for all .
We now design a vertex partition to make approximately constant on most cells. For each , we partition into cells on which (viewed as a function from to ) only fluctuates by , plus an exceptional cell of size coming from the values where is excessively large (larger than ). Combining all these partitions together, we can write for some , where has cardinality at most , and for all , the eigenfunctions all fluctuate by at most . In particular, if , then (by (4) and (6)) the entries of fluctuate by at most on each block . If we let be the mean value of these entries on , we thus have
for any and , where we view the indicator functions as column vectors of dimension .
for all pairs outside of an exceptional set with
for any , by (10) and the Cauchy-Schwarz inequality.
for any .
Let be the set of all pairs where either , , , or
for all . The left-hand side is just . As , we have
and so (since )
To prove Lemma 1, one argues similarly (after modifying as necessary), except that the initial partition of constructed above needs to be subdivided further into equitable components (of size ), plus some remainder sets which can be aggregated into an exceptional component of size (and which can then be redistributed amongst the other components to arrive at a truly equitable partition). We omit the details.
Remark 1 It is easy to verify that needs to be growing exponentially in in order for the above argument to work, which leads to tower-exponential bounds in the number of cells in the partition. It was shown by Gowers that a tower-exponential bound is actually necessary here. By varying , one basically obtains the strong regularity lemma first established by Alon, Fischer, Krivelevich, and Szegedy; in the opposite direction, setting essentially gives the weak regularity lemma of Frieze and Kannan.
Remark 2 If we specialise to a Cayley graph, in which is a finite abelian group and for some (symmetric) subset of , then the eigenvectors are characters, and one essentially recovers the arithmetic regularity lemma of Green, in which the vertex partition classes are given by Bohr sets (and one can then place additional regularity properties on these Bohr sets with some additional arguments). The components of , representing high, medium, and low eigenvalues of , then become a decomposition associated to high, medium, and low Fourier coefficients of .
Remark 3 The use of spectral theory here is parallel to the use of Fourier analysis to establish results such as Roth’s theorem on arithmetic progressions of length three. In analogy with this, one could view hypergraph regularity as being a sort of “higher order spectral theory”, although this spectral perspective is not as convenient as it is in the graph case.
Van Vu and I have just uploaded to the arXiv our paper “Random matrices: Universality of eigenvectors“, submitted to Random Matrices: Theory and Applications. This paper concerns an extension of our four moment theorem for eigenvalues. Roughly speaking, that four moment theorem asserts (under mild decay conditions on the coefficients of the random matrix) that the fine-scale structure of individual eigenvalues of a Wigner random matrix depend only on the first four moments of each of the entries.
In this paper, we extend this result from eigenvalues to eigenvectors, and specifically to the coefficients of, say, the eigenvector of a Wigner random matrix . Roughly speaking, the main result is that the distribution of these coefficients also only depends on the first four moments of each of the entries. In particular, as the distribution of coefficients eigenvectors of invariant ensembles such as GOE or GUE are known to be asymptotically gaussian real (in the GOE case) or gaussian complex (in the GUE case), the same asymptotic automatically holds for Wigner matrices whose coefficients match GOE or GUE to fourth order.
(A technical point here: strictly speaking, the eigenvectors are only determined up to a phase, even when the eigenvalues are simple. So, to phrase the question properly, one has to perform some sort of normalisation, for instance by working with the coefficients of the spectral projection operators instead of the eigenvectors, or rotating each eigenvector by a random phase, or by fixing the first component of each eigenvector to be positive real. This is a fairly minor technical issue here, though, and will not be discussed further.)
This theorem strengthens a four moment theorem for eigenvectors recently established by Knowles and Yin (by a somewhat different method), in that the hypotheses are weaker (no level repulsion assumption is required, and the matrix entries only need to obey a finite moment condition rather than an exponential decay condition), and a slightly stronger conclusion (less regularity is needed on the test function, and one can handle the joint distribution of polynomially many coefficients, rather than boundedly many coefficients). On the other hand, the Knowles-Yin paper can also handle generalised Wigner ensembles in which the variances of the entries are allowed to fluctuate somewhat.
The method used here is a variation of that in our original paper (incorporating the subsequent improvements to extend the four moment theorem from the bulk to the edge, and to replace exponential decay by a finite moment condition). That method was ultimately based on the observation that if one swapped a single entry (and its adjoint) in a Wigner random matrix, then an individual eigenvalue would not fluctuate much as a consequence (as long as one had already truncated away the event of an unexpectedly small eigenvalue gap). The same analysis shows that the projection matrices obeys the same stability property.
As an application of the eigenvalue four moment theorem, we establish a four moment theorem for the coefficients of resolvent matrices , even when is on the real axis (though in that case we need to make a level repulsion hypothesis, which has been already verified in many important special cases and is likely to be true in general). This improves on an earlier four moment theorem for resolvents of Erdos, Yau, and Yin, which required to stay some distance away from the real axis (specifically, that for some small ).