The Brown measure of the sum of a self-adjoint element and an elliptic element

Ho, Ching-Wei

doi:10.48550/arxiv.2007.06100

Cited by 3 publications

(17 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The following theorem is established in Theorem 5.4 and Corollary 5.7. This result extends Theorems 3.7 and 4.1 in [15] to possibly-unbounded x 0 affiliated with A .…”

Section: Consider the Functionsupporting

confidence: 81%

“…In this section, we study the Brown measure of x 0 + c α,β , where x 0 ∈ A ∆ is self-adjoint, and α ≥ 0 (the case α = 0 agrees with the one computed in Theorem 4.5). The computation in [15] relies only on the computations of holomorphic functions; the compactness of the support of µ x0 in [15] does not play a role (since the computation of the holomorphic maps can be restricted to a Stolz angle). The Brown measure computation of [15] can be automatically carried over to the unbounded self-adjoint x 0 .…”

Section: The Brown Measure Computationmentioning

confidence: 99%

“…In particular, x 0 is a bounded random variable. The author computed the Brown measure of x 0 + c α,β in [15]. In this paper, we extend the computation to an unbounded self-adjoint random variable x 0 affiliated with A .…”

Section: Introductionmentioning

confidence: 99%

“…Later, Hall and the author [13] computed the Brown measure of the sum of an arbitrary bounded self-adjoint random variable x 0 and an imaginary multiple of semicircular variable σ t , freely independent from x 0 . Using the results in [13], the author [15] computed the Brown measure of the sum of a bounded self-adjoint random variable x 0 and an elliptic variable c α,β that is freely independent from x 0 . The results in [13,15,16] assume the boundedness of x 0 in the process of computing the PDE.…”

Section: Introductionmentioning

confidence: 99%

“…Using the results in [13], the author [15] computed the Brown measure of the sum of a bounded self-adjoint random variable x 0 and an elliptic variable c α,β that is freely independent from x 0 . The results in [13,15,16] assume the boundedness of x 0 in the process of computing the PDE. The key observation in this paper is to show that the function S(t, λ, ε) = τ [log((x 0 + iσ t − λ) * (x 0 + iσ t − λ) + ε)], t > 0, λ ∈ C, ε > 0 (1.1) satisfies the same PDE derived in [13] when x 0 is an unbounded random variable affiliated with the W * -probability space A , using noncommutative integration theory.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

The Brown measure of unbounded variables with free semicircular imaginary part

Ho¹

2020

Preprint

Self Cite

View full text Add to dashboard Cite

Let x0 be a possibly-unbounded self-adjoint random variable, σα and σ β are semicircular variables with variances α ≥ 0 and β > 0 respectively (when α = 0, σα = 0). Suppose x0, σα, and σβ are all freely independent. We compute the Brown measure of x0 + σα + iσ β , extending the previous computations which only work for bounded self-adjoint random variable x0. The Brown measure in this unbounded case has the same structure as in the bounded case; it has connections to the free convolution x0 + σ α+β . We also compute the example where x0 is Cauchy-distributed.

show abstract

“…The following theorem is established in Theorem 5.4 and Corollary 5.7. This result extends Theorems 3.7 and 4.1 in [15] to possibly-unbounded x 0 affiliated with A .…”

Section: Consider the Functionsupporting

confidence: 81%

Section: The Brown Measure Computationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The Brown measure of unbounded variables with free semicircular imaginary part

Ho¹

2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Brown measure of the sum of an elliptic operator and a free random variable in a finite von Neumann algebra

Zhong¹

2021

Preprint

View full text Add to dashboard Cite

We obtain, under mild assumptions, a formula for the Brown measure of the sum of a twisted elliptic operator g t,γ with an arbitrary random variable x 0 , freely independent from g t,γ . The twisted elliptic operators include Voiculescu's circular operator and elliptic operators. We show that the Brown measure of the sum of a twisted elliptic operator with a free random variable is the push-forward measure of the Brown measures of the sum of a free circular operator with the same free random variable under a natural map, provided that this map is oneto-one and non-singular. This generalizes earlier results about free additive Brownian motions where the free random variable x 0 is assumed to be selfadjoint. In particular, we prove that the Brown measure of the sum of a Haar unitary operator and a twisted elliptic element is supported in a deformed ring where the inner boundary is a circle and the outer boundary is an ellipse, and its density is constant along a family of ellipses. The approach is based on a Hermitian reduction and subordination functions.

show abstract

The heat flow conjecture for random matrices

Hall¹,

Ho²

2022

Preprint

Self Cite

View full text Add to dashboard Cite

Recent results by various authors have established a "model deformation phenomenon" in random matrix theory. Specifically, it is possible to construct pairs of random matrix models such that the limiting eigenvalue distributions are connected by push-forward under an explicitly constructible map of the plane to itself. In this paper, we argue that the analogous transformation at the finite-N level can be accomplished by applying an appropriate heat flow to the characteristic polynomial of the first model.We develop several conjectures of the following sort. We find certain pairs of random matrix models and we apply a certain heat-type operator to the characteristic polynomial p of the first model, giving a new polynomial q. We then conjecture that, when N is large, the zeros of q have the same bulk distribution as the eigenvalues of the second random matrix model. As a special case, suppose we apply the standard heat operator for time 1/N to the characteristic polynomial p of an N × N GUE matrix, giving a new polynomial q. Then we conjecture that the zeros of q will be asymptotically uniformly distributed over the unit disk.At a more refined level, we conjecture that as the characteristic polynomial of the first model evolves under the appropriate heat flow, its zeros will evolve along the characteristic curves of a certain PDE. Our conjectures are supported two rigorous results: a deformation theorem for the second moments of the characteristic polynomial and a PDE satisfied by the log potential of the zeros of a heat-evolving polynomial. Contents 1. The model deformation phenomenon in random matrix theory 2. The heat flow conjectures 2.1. The heat flow conjectures relating the circular and semicircular laws 2.2. The general additive heat flow conjecture 2.3. The general multiplicative heat flow conjecture 3. The deformation theorem for second moments of the characteristic polynomial 3.1. The second moment 3.2. Additive case 3.3. Multiplicative case 4. The PDE perspective 4.1. The PDEs for the Brown measure 4.2. The PDEs for T N and σ N 4.3. A PDE argument for the conjectures References

show abstract

The Brown measure of the sum of a self-adjoint element and an elliptic element

Cited by 3 publications

References 36 publications

The Brown measure of unbounded variables with free semicircular imaginary part

The Brown measure of unbounded variables with free semicircular imaginary part

Brown measure of the sum of an elliptic operator and a free random variable in a finite von Neumann algebra

The heat flow conjecture for random matrices

Contact Info

Product

Resources

About