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

Zhong, Ping

doi:10.48550/arxiv.2108.09844

Cited by 4 publications

(9 citation statements)

References 39 publications

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“…They also show that all the Brown measures obtained by varying τ with s and u fixed are related. (4) The paper [27] of Zhong considers the additive counterpart z s,τ of b s,τ and uses free probability to compute the Brown measure of x 0 + z s,τ , where x 0 is freely independent of z s,τ . In the case that x 0 is Hermitian, Zhong relates all the Brown measures obtained by varying τ with s and x 0 fixed.…”

Section: The Model Deformation Phenomenon In Random Matrix Theorymentioning

confidence: 99%

“…The circular-to-semicircular conjecture is the case X N 0 = 0, τ 0 = 1, and τ = 0, while the semicircular-to-circular conjecture is the case X N 0 = 0, τ 0 = 0, and τ = 1. The limiting eigenvalue distribution of X N 0 + Z N s,τ , meanwhile, has been computed by Zhong [27] using methods of free probability. See also similar results in [15] obtained by the PDE method.…”

Section: 2mentioning

confidence: 99%

“…The maps in [27], meanwhile, have the "push-forward property," meaning that pushing forward under such a map transforms the limiting eigenvalue distribution of X N 0 + Z N s,τ for τ = s in to the limiting eigenvalue distribution of X N 0 + Z N s,τ for an arbitrary value of τ. By composing one such map with the inverse of another, we reach the following conclusion.…”

Section: 2mentioning

confidence: 99%

“…Conclusion 2.10. Conjecture 2.9 can be restated as saying that z s,τ0 j (τ ) should be approximately equal to Φ s,τ0,τ (z j (τ 0 )), where according to results of Zhong [27], pushing forward under Φ s,τ0,τ transforms the limiting eigenvalue distribution of X N 0 + Z N s,τ0 into the limiting eigenvalue distribution of X N 0 + Z N s,τ . As explained in Section 4.1, we can also understand the right-hand side of (2.…”

Section: 2mentioning

confidence: 99%

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The heat flow conjecture for random matrices

Hall¹,

Ho²

2022

Preprint

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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

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Section: The Model Deformation Phenomenon In Random Matrix Theorymentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

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The heat flow conjecture for random matrices

Hall¹,

Ho²

2022

Preprint

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“…In [24], the author calculated the Brown measure density of g t + u. It is showed that the Brown measure of g t + u is the pushforward measure of c t + u under some natural pushforward map.…”

Section: Some Miscellaneous Applicationsmentioning

confidence: 99%

Brown measure of $R$-diagonal operators, revisited

Zhong¹

2022

Preprint

Self Cite

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We observe that subordination functions were used implicitly in Haagerup-Schultz's approach for the Brown measure of R-diagonal operators. This allows us to simplify the original argument and find a connection with the other approach due to Belinschi-Śniady-Speicher. The Brown measure formula can be rewritten in terms of subordination functions. * D |T +|λ|U |. Hence, for any λ ∈ C\{0} fixed, the distribution of |T − λ1| can be calculated as addition of two R-diagonal operators. The problem is then reduced to understand the free additive convolution of a symmetric probability measure and a Bernoulli distribution. The above observations are the key to find a formula for the Fuglede-Kadison determinant of T − λ1.

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Wegner estimate and upper bound on the eigenvalue condition number of non‐Hermitian random matrices

Erdős,

2024

Comm Pure Appl Math

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We consider non‐Hermitian random matrices of the form , where is a general deterministic matrix and consists of independent entries with zero mean, unit variance, and bounded densities. For this ensemble, we prove (i) a Wegner estimate, that is, that the local density of eigenvalues is bounded by and (ii) that the expected condition number of any bulk eigenvalue is bounded by ; both results are optimal up to the factor . The latter result complements the very recent matching lower bound obtained by Cipolloni et al. and improves the ‐dependence of the upper bounds by Banks et al. and Jain et al. Our main ingredient, a near‐optimal lower tail estimate for the small singular values of , is of independent interest.

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

Cited by 4 publications

References 39 publications

The heat flow conjecture for random matrices

The heat flow conjecture for random matrices

Brown measure of $R$-diagonal operators, revisited

Wegner estimate and upper bound on the eigenvalue condition number of non‐Hermitian random matrices

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