TY - JOUR
AB - We investigate the free boundary Schur process, a variant of the Schur process introduced by Okounkov and Reshetikhin, where we allow the first and the last partitions to be arbitrary (instead of empty in the original setting). The pfaffian Schur process, previously studied by several authors, is recovered when just one of the boundary partitions is left free. We compute the correlation functions of the process in all generality via the free fermion formalism, which we extend with the thorough treatment of “free boundary states.” For the case of one free boundary, our approach yields a new proof that the process is pfaffian. For the case of two free boundaries, we find that the process is not pfaffian, but a closely related process is. We also study three different applications of the Schur process with one free boundary: fluctuations of symmetrized last passage percolation models, limit shapes and processes for symmetric plane partitions and for plane overpartitions.
AU - Betea, Dan
AU - Bouttier, Jeremie
AU - Nejjar, Peter
AU - Vuletic, Mirjana
ID - 556
IS - 12
JF - Annales Henri Poincare
SN - 14240637
TI - The free boundary Schur process and applications I
VL - 19
ER -
TY - JOUR
AB - We consider large random matrices X with centered, independent entries which have comparable but not necessarily identical variances. Girko's circular law asserts that the spectrum is supported in a disk and in case of identical variances, the limiting density is uniform. In this special case, the local circular law by Bourgade et. al. [11,12] shows that the empirical density converges even locally on scales slightly above the typical eigenvalue spacing. In the general case, the limiting density is typically inhomogeneous and it is obtained via solving a system of deterministic equations. Our main result is the local inhomogeneous circular law in the bulk spectrum on the optimal scale for a general variance profile of the entries of X.
AU - Alt, Johannes
AU - Erdös, László
AU - Krüger, Torben H
ID - 566
IS - 1
JF - Annals Applied Probability
TI - Local inhomogeneous circular law
VL - 28
ER -
TY - JOUR
AB - We consider a Wigner-type ensemble, i.e. large hermitian N×N random matrices H=H∗ with centered independent entries and with a general matrix of variances Sxy=𝔼∣∣Hxy∣∣2. The norm of H is asymptotically given by the maximum of the support of the self-consistent density of states. We establish a bound on this maximum in terms of norms of powers of S that substantially improves the earlier bound 2∥S∥1/2∞ given in [O. Ajanki, L. Erdős and T. Krüger, Universality for general Wigner-type matrices, Prob. Theor. Rel. Fields169 (2017) 667–727]. The key element of the proof is an effective Markov chain approximation for the contributions of the weighted Dyck paths appearing in the iterative solution of the corresponding Dyson equation.
AU - Erdös, László
AU - Mühlbacher, Peter
ID - 5971
JF - Random matrices: Theory and applications
SN - 2010-3263
TI - Bounds on the norm of Wigner-type random matrices
ER -
TY - GEN
AB - We study the unique solution $m$ of the Dyson equation \[ -m(z)^{-1} = z - a
+ S[m(z)] \] on a von Neumann algebra $\mathcal{A}$ with the constraint
$\mathrm{Im}\,m\geq 0$. Here, $z$ lies in the complex upper half-plane, $a$ is
a self-adjoint element of $\mathcal{A}$ and $S$ is a positivity-preserving
linear operator on $\mathcal{A}$. We show that $m$ is the Stieltjes transform
of a compactly supported $\mathcal{A}$-valued measure on $\mathbb{R}$. Under
suitable assumptions, we establish that this measure has a uniformly
$1/3$-H\"{o}lder continuous density with respect to the Lebesgue measure, which
is supported on finitely many intervals, called bands. In fact, the density is
analytic inside the bands with a square-root growth at the edges and internal
cubic root cusps whenever the gap between two bands vanishes. The shape of
these singularities is universal and no other singularity may occur. We give a
precise asymptotic description of $m$ near the singular points. These
asymptotics generalize the analysis at the regular edges given in the companion
paper on the Tracy-Widom universality for the edge eigenvalue statistics for
correlated random matrices [arXiv:1804.07744] and they play a key role in the
proof of the Pearcey universality at the cusp for Wigner-type matrices
[arXiv:1809.03971,arXiv:1811.04055]. We also extend the finite dimensional band
mass formula from [arXiv:1804.07744] to the von Neumann algebra setting by
showing that the spectral mass of the bands is topologically rigid under
deformations and we conclude that these masses are quantized in some important
cases.
AU - Alt, Johannes
AU - Erdös, László
AU - Krüger, Torben H
ID - 6183
T2 - arXiv
TI - The Dyson equation with linear self-energy: Spectral bands, edges and cusps
ER -
TY - JOUR
AB - We consider spectral properties and the edge universality of sparse random matrices, the class of random matrices that includes the adjacency matrices of the Erdős–Rényi graph model G(N, p). We prove a local law for the eigenvalue density up to the spectral edges. Under a suitable condition on the sparsity, we also prove that the rescaled extremal eigenvalues exhibit GOE Tracy–Widom fluctuations if a deterministic shift of the spectral edge due to the sparsity is included. For the adjacency matrix of the Erdős–Rényi graph this establishes the Tracy–Widom fluctuations of the second largest eigenvalue when p is much larger than N−2/3 with a deterministic shift of order (Np)−1.
AU - Lee, Jii
AU - Schnelli, Kevin
ID - 690
IS - 1-2
JF - Probability Theory and Related Fields
TI - Local law and Tracy–Widom limit for sparse random matrices
VL - 171
ER -