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Tokyo Probability Seminar
Seminar information archive ～05/28｜Next seminar｜Future seminars 05/29～
| Date, time & place | Monday 16:00 - 17:30 126Room #126 (Graduate School of Math. Sci. Bldg.) |
|---|---|
| Organizer(s) | Makiko Sasada, Shuta Nakajima, Masato Hoshino |
2025/01/20
16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)
We are having teatime from 15:15 in the common room on the second floor. Please join us.
Arka Adhikari (University of Maryland)
Spectral measure for uniform d-regular digraphs
We are having teatime from 15:15 in the common room on the second floor. Please join us.
Arka Adhikari (University of Maryland)
Spectral measure for uniform d-regular digraphs
[ Abstract ]
Consider the matrix \sfA\GG chosen uniformly at random from the finite
set of all N-dimensional matrices of zero main-diagonal and binary entries,
having each row and column of \sfA\GG sum to d.
That is, the adjacency matrix for the uniformly random
d-regular simple digraph \GG. Fixing d≥3, it has long been conjectured
that as N→∞ the corresponding empirical eigenvalue distributions converge
weakly, in probability, to an explicit non-random limit,
given by the Brown measure of the free sum of d Haar unitary operators.
We reduce this conjecture to bounding the decay in N of the probability that
the minimal singular value of the shifted matrix \sfA(w)=\sfA\GG−w\sfI
is very small. While the latter remains a challenging task, the required bound is
comparable to the recently established control on the singularity of \sfA\GG.
The reduction is achieved here by sharp estimates
on the behavior at large N, near the real line, of the Green's function (aka resolvent)
of the Hermitization of \sfA(w), which is of independent interest.
Joint w/ A. Dembo
Consider the matrix \sfA\GG chosen uniformly at random from the finite
set of all N-dimensional matrices of zero main-diagonal and binary entries,
having each row and column of \sfA\GG sum to d.
That is, the adjacency matrix for the uniformly random
d-regular simple digraph \GG. Fixing d≥3, it has long been conjectured
that as N→∞ the corresponding empirical eigenvalue distributions converge
weakly, in probability, to an explicit non-random limit,
given by the Brown measure of the free sum of d Haar unitary operators.
We reduce this conjecture to bounding the decay in N of the probability that
the minimal singular value of the shifted matrix \sfA(w)=\sfA\GG−w\sfI
is very small. While the latter remains a challenging task, the required bound is
comparable to the recently established control on the singularity of \sfA\GG.
The reduction is achieved here by sharp estimates
on the behavior at large N, near the real line, of the Green's function (aka resolvent)
of the Hermitization of \sfA(w), which is of independent interest.
Joint w/ A. Dembo
