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Sums of GUE matrices and focus of hives from correlation decay of eigengaps


Hariharan Narayanan, Scott Sheffield, and I’ve simply uploaded to the arXiv our paper “Sums of GUE matrices and focus of hives from correlation decay of eigengaps“. It is a personally satisfying paper for me, because it connects the work I did as a graduate pupil (with Allen Knutson and Chris Woodward) on sums of Hermitian matrices, with newer work I did (with Van Vu) on random matrix idea, in addition to a number of different outcomes by different authors scattered throughout varied mathematical subfields.

Suppose {A, B} are two {n times n} Hermitian matrices with eigenvalues {lambda = (lambda_1,dots,lambda_n)} and {mu = (mu_1,dots,mu_n)} respectively (organized in non-increasing order. What can one say in regards to the eigenvalues {nu = (nu_1,dots,nu_n)} of the sum {A+B}? There at the moment are some ways to reply this query exactly; considered one of them, launched by Allen and myself a few years in the past, is that there exists a sure triangular array of numbers referred to as a “hive” that has {lambda, mu, nu} as its boundary values. However, by the pioneering work of Voiculescu in free chance, we all know within the massive {n} restrict that if {lambda, mu} are asymptotically drawn from some limiting distribution, and {A} and {B} are drawn independently at random (utilizing the unitarily invariant Haar measure) amongst all Hermitian matrices with the indicated eigenvalues, then (below gentle hypotheses on the distribution, and below appropriate normalization), {nu} will virtually certainly have a limiting distribution that’s the free convolution of the 2 unique distributions.

One in every of my favorite open issues is to give you a idea of “free hives” that enables one to clarify the latter reality from the previous. That is nonetheless unresolved, however we at the moment are starting to make a little bit of progress in the direction of this purpose. We all know (as an illustration from the calculations of Coquereaux and Zuber) that if {A, B} are drawn independently at random with eigenvalues {lambda, mu}, then the eigenvalues {nu} of {A+B} are distributed in accordance with the boundary values of an “augmented hive” with two boundaries {lambda,mu}, drawn uniformly at random from the polytope of all such augmented hives. (This augmented hive is principally an everyday hive with one other kind of sample, specifically a Gelfand-Tsetlin sample, glued to at least one facet of it.) So, if one might present some kind of focus of measure for the entries of this augmented hive, and calculate what these entries concentrated to, one ought to presumably have the ability to recuperate Voiculescu’s consequence after some calculation.

On this paper, we’re in a position to accomplish the primary half of this purpose, assuming that the spectra {lambda, mu} are usually not deterministic, however somewhat drawn from the spectra of rescaled GUE matrices (thus {A,B} are unbiased rescaled copies of the GUE ensemble). We’ve chosen to normalize issues in order that the eigenvalues {lambda,mu} have dimension {O(n)}, in order that the entries of the augmented hive have entries {O(n^2)}. Our result’s then that the entries of the augmented hive in truth have a typical deviation of {o(n^2)}, thus exhibiting just a little little bit of focus. (Really, from the Brunn-Minkowski inequality, the distribution of those entries is log concave, so as soon as as soon as controls the usual deviation one additionally will get a little bit of exponential decay past the usual deviation; Narayanan and Sheffield had additionally not too long ago established the existence of a charge operate for this kind of mannequin.) Presumably one ought to get a lot better focus, and one ought to have the ability to deal with different fashions than the GUE ensemble, however that is the primary advance that we had been in a position to obtain.

Augmented hives appear tough to work with straight, however by adapting the octahedron recurrence launched for this downside by Knutson, Woodward, and myself a while in the past (which is said to the associativity {(A+B)+C = A+(B+C)} of addition for Hermitian matrices), one can assemble a piecewise linear volume-preserving map between the cone of augmented hives, and the product of two Gelfand-Tsetlin cones. The issue then reduces to establishing focus of measure for sure piecewise linear maps on merchandise of Gelfand-Tsetlin cones (endowed with a sure GUE-type measure). It is a promising formulation as a result of Gelfand-Tsetlin cones are by now fairly effectively understood.

However, the piecewise linear map, initially outlined by iterating the octahedron relation {f = max(a+c,b+d)-e}, seems considerably daunting. Thankfully, there may be an specific formulation of this map because of Speyer, because the supremum of sure linear maps related to good matchings of a sure “excavation graph”. For us it was handy to work with the twin of this excavation graph, and affiliate these linear maps to sure “lozenge tilings” of a hexagon.

It could be extra handy to review the focus of every linear map individually, somewhat than their supremum. By the Cheeger inequality, it seems that one can relate the latter to the previous supplied that one has good management on the Cheeger fixed of the underlying measure on the Gelfand-Tsetlin cones. Thankfully, the measure is log-concave, so one can use the very current work of Klartag on the KLS conjecture to remove the supremum (as much as a logarithmic loss which is just reasonably annoying to take care of).

It stays to acquire focus on the linear map related to a given lozenge tiling. After stripping away some contributions coming from lozenges close to the sting (utilizing some eigenvalue rigidity outcomes of Van Vu and myself), one is left with some bulk contributions which in the end contain eigenvalue interlacing gaps resembling

displaystyle  lambda_i - lambda_{n-1,i}

the place {lambda_{n-1,i}} is the {i^{th}} eigenvalue of the highest left {n-1 times n-1} minor of {A}, and {i} is within the bulk area {varepsilon n leq i leq (1-varepsilon) n} for some fastened {varepsilon > 0}. To get the specified consequence, one wants some non-trivial correlation decay in {i} for these statistics. If one was working with eigenvalue gaps {lambda_i - lambda_{i+1}} somewhat than interlacing outcomes, then such correlation decay was conveniently obtained for us by current work of Cippoloni, Erdös, and Schröder. So the final remaining problem is to grasp the relation between eigenvalue gaps and interlacing gaps.

For this we turned to the work of Metcalfe, who uncovered a determinantal course of construction to this downside, with a kernel related to Lagrange interpolation polynomials. It’s attainable to satisfactorily estimate varied integrals of those kernels utilizing the residue theorem and eigenvalue rigidity estimates, thus finishing the required evaluation.

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