On the operator norm of non-commutative polynomials in deterministic matrices and iid Haar unitary matrices

Parraud, Félix

doi:10.1007/s00440-021-01101-0

Cited by 6 publications

(15 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We have now defined the non-commutative differential of the exponential of a polynomial twice, in ( 8) and ( 9). However those two definitions are compatible thanks to the following proposition (see [9], Proposition 2.2 for the proof).…”

Section: Non-commutative Polynomials and Derivativesmentioning

confidence: 95%

Asymptotic expansion of smooth functions in polynomials in deterministic matrices and iid GUE matrices

Parraud¹

2020

Preprint

View full text Add to dashboard Cite

Let X N be a family of N × N independent GUE random matrices, Z N a family of deterministic matrices, P a self-adjoint non-commutative polynomial, that is for any N , P (X N ) is self-adjoint, f a smooth function. We prove that for any k, if f is smooth enough, there exist deterministic constantsBesides the constants α P i (f, Z N ) are built explicitly with the help of free probability. In particular, if x is a free semicircular system, then when the support of f and the spectrum of P (x, Z N ) are disjoint, for any i, α P i (f, Z N ) = 0. As a corollary, we prove that given α < 1/2, for N large enough, every eigenvalue of P (X N , Z N ) is N −α -close from the spectrum of P (x, Z N ).

show abstract

Section: Non-commutative Polynomials and Derivativesmentioning

confidence: 95%

Asymptotic expansion of smooth functions in polynomials in deterministic matrices and iid GUE matrices

Parraud¹

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Thus if one can show that the right-hand side of this inequality converges towards zero when N goes to infinity, then asymptotically there is no eigenvalue in the segment [a, b]. We did prove so in [37] where we showed that given a smooth function f , there is a constant α P 0 (f ), which can be computed explicitly with the help of free probability, such that…”

Section: Introductionmentioning

confidence: 90%

“…This approach was refined in [36] where we proved an asymptotic expansion for polynomials in independent GUE matrices and deterministic matrices. In [37], we used the heuristic of [18] to study polynomials of independent Haar unitary matrices and deterministic matrices. Thus by combining the different tools used in those papers, we prove an asymptotic expansion in the unitary case.…”

Section: Introductionmentioning

confidence: 99%

“…Since there exist no such formula for the eigenvalues of a polynomial in independent Haar unitary matrices, we cannot adapt this proof. Instead we rely on the strategy developed in [18,37,36]. The main idea is to interpolate Haar unitary matrices and free Haar unitaries with the help of a free unitary Brownian motion.…”

Section: Introductionmentioning

confidence: 99%

“…The main idea is to interpolate Haar unitary matrices and free Haar unitaries with the help of a free unitary Brownian motion. The main tool to do so is free stochastic calculus, although in this paper we rely on Proposition 3.3 of [37] to circumvent most of those computations. Once they are done, we are left with Equation (16), which is strongly reminiscent of the Schwinger-Dyson Equation for the unitary group.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Asymptotic Expansion of Smooth Functions in Polynomials in Deterministic Matrices and iid GUE Matrices

Parraud

2022

Commun. Math. Phys.

Self Cite

View full text Add to dashboard Cite

Let U N be a family of N × N independent Haar unitary random matrices and their adjoints, Z N a family of deterministic matrices, P a self-adjoint noncommutative polynomial, i.e. such that for any N , P (U N , Z N ) is self-adjoint, f a smooth function. We prove that for any k, if f is smooth enough, there exist deterministic constantsBesides the constants α P i (f, Z N ) are built explicitly with the help of free probability. As a corollary, we prove that given α < 1/2, for N large enough, every eigenvalue of P (U N , Z N ) is N −α -close to the spectrum of P (u, Z N ) where u is a d-tuple of free Haar unitaries. We also prove the convergence of the norm of any polynomial P (U N ⊗ IM , IN ⊗ Y M ) as long as the family Y M converges strongly and that M ≪ N ln −5/2 (N ).

show abstract

Convergence for noncommutative rational functions evaluated in random matrices

et al. 2022

View full text Add to dashboard Cite

On the operator norm of non-commutative polynomials in deterministic matrices and iid Haar unitary matrices

Cited by 6 publications

References 24 publications

Asymptotic expansion of smooth functions in polynomials in deterministic matrices and iid GUE matrices

Asymptotic expansion of smooth functions in polynomials in deterministic matrices and iid GUE matrices

Asymptotic Expansion of Smooth Functions in Polynomials in Deterministic Matrices and iid GUE Matrices

Convergence for noncommutative rational functions evaluated in random matrices

Contact Info

Product

Resources

About