Proves Cosine and Hermite universality conjectures for roots of derivatives of even entire functions with real roots and for Jensen polynomials by establishing finite free probability limit theorems for repeated differentiation.
Marcus, Polynomial convolutions and (finite) free probability, arXiv:2108.07054 [math.CO]
6 Pith papers cite this work. Polarity classification is still indexing.
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Defines (n,d)-rectangular cumulants that linearize (n,d)-rectangular convolution in finite free probability and converge to q-rectangular free cumulants as d→∞ with 1+n/d→q.
Limiting root distribution after repeated fractional differentiation is the push-forward of the initial distribution under a characteristic flow of the log-potential PDE.
The finite R-transform of a polynomial differs from the Voiculescu R-transform of its empirical root distribution by O(N^{-1}), providing an analytic proof that finite free additive convolution converges to free additive convolution.
t-deformed convolution and cumulants on formal power series yield LLN and CLT analogues that recover classical convolution at t=-1 and finite free generators at t=d, with explicit infinitesimal generators for the associated semigroups.
Computational discovery via FlowBoost supports conjectures on the singular values of the coupling matrix E_n being 2^{-k/2} independent of n, a sharp p=2 critical exponent for p-Stam inequalities, and bifurcation of extremals for p<2.
citing papers explorer
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Universality for roots of derivatives of entire functions via finite free probability
Proves Cosine and Hermite universality conjectures for roots of derivatives of even entire functions with real roots and for Jensen polynomials by establishing finite free probability limit theorems for repeated differentiation.
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Cumulants in rectangular finite free probability and beta-deformed singular values
Defines (n,d)-rectangular cumulants that linearize (n,d)-rectangular convolution in finite free probability and converge to q-rectangular free cumulants as d→∞ with 1+n/d→q.
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Roots of polynomials under repeated differentiation and repeated applications of fractional differential operators
Limiting root distribution after repeated fractional differentiation is the push-forward of the initial distribution under a characteristic flow of the log-potential PDE.
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An analytic approach to the finite R-transform
The finite R-transform of a polynomial differs from the Voiculescu R-transform of its empirical root distribution by O(N^{-1}), providing an analytic proof that finite free additive convolution converges to free additive convolution.
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Convolution, cumulants and infinitesimal generators in the formal power series ring
t-deformed convolution and cumulants on formal power series yield LLN and CLT analogues that recover classical convolution at t=-1 and finite free generators at t=d, with explicit infinitesimal generators for the associated semigroups.
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Spectral Structure in Finite Free Information Inequalities and $p$-Stam Phase Transitions
Computational discovery via FlowBoost supports conjectures on the singular values of the coupling matrix E_n being 2^{-k/2} independent of n, a sharp p=2 critical exponent for p-Stam inequalities, and bifurcation of extremals for p<2.