Heat Kernel Empirical Laws on $${\mathbb {U}}_N$$ U N and $${\mathbb {GL}}_N$$ GL N

Kemp, Todd

doi:10.1007/s10959-015-0643-7

Cited by 12 publications

(3 citation statements)

References 37 publications

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Section: Fitting Hyperplanes In D Dimensionsmentioning

confidence: 99%

“…By standard Fourier arguments (e.g. §2.2, [20]), solutions to the time-dependent diffusion equation for the probability density of p ( t ) of the unitary Brownian motion U t satisfy, for some Poincaré-like constant C n > 0 depending only on n , with d the L 2 metric and ν the uniform (Haar) measure on the unitary group 𝕌( n ). Therefore, if r ( t ) satisfies: the RHS in (S59) vanishes in large t , and global exponential stability is assured.…”

Section: Background On Random Variablesmentioning

confidence: 99%

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Stochastic Voronoi Tessellations as Models for Cellular Neighborhoods in Simple Multicellular Organisms

Srinivasan,

Höhn,

Goldstein

2024

Preprint

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Recent work on distinct multicellular organisms has revealed a hitherto unknown type of biological noise; rather than a regular arrangement, cellular neighborhood volumes, obtained by Voronoi tessellations of the cell locations, are broadly distributed and consistent with gamma distributions. We propose an explanation for those observations in the case of the algaVolvox, whose somatic cells are embedded in an extracellular matrix (ECM) they export. Both a solvable one-dimensional model of ECM growth derived from bursty transcriptional activity and a two-dimensional "Voronoi liquid" model are shown to provide one-parameter families that smoothly interpolate between the empirically-observed near-maximum-entropy gamma distributions and the crystalline limit of Gaussian distributions governed by the central limit theorem. These results highlight a universal consequence of intrinsic biological noise on the architecture of certain tissues.

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Section: Fitting Hyperplanes In D Dimensionsmentioning

confidence: 99%

Section: Background On Random Variablesmentioning

confidence: 99%

Stochastic Voronoi Tessellations as Models for Cellular Neighborhoods in Simple Multicellular Organisms

Srinivasan,

Höhn,

Goldstein

2024

Preprint

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“…Here we recall the construction of a certain space of non-commutative functions given as L 2 -limits of trace polynomials, uniformly over all noncommutative laws. Trace polynomials have been studied in many previous works such as [82,78,83,81,86,11,22,44,46,47,18]. The uniform L 2 -completion of trace polynomials was first introduced in [40,41,43], and its relationship with continuous model theory was addressed in [42, §3.5].…”

Section: 2mentioning

confidence: 99%

A random matrix approach to the Peterson-Thom conjecture

Hayes¹

2022

Indiana Univ. Math. J.

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In this paper we show various new structural properties of free group factors using the recent resolution (due independently to Belinschi-Capitaine and Bordenave-Collins) of the Peterson-Thom conjecture. These results include the resolution to the coarseness conjecture independently due to the first-named author and Popa, a generalization of Ozawa-Popa's celebrated strong solidity result using vastly more general versions of the normalizer (and in an ultraproduct setting), a dichotomy result for intertwining of maximal amenable subalgebras of interpolated free group factors, as well as application to ultraproduct embeddings of nonamenable subalgebras of interpolated free group factors.

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Coherent States for Compact Lie Groups and Their Large-N Limits

Hall¹

2018

Springer Proceedings in Physics

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The first two parts of this article surveys results related to the heatkernel coherent states for a compact Lie group K. I begin by reviewing the definition of the coherent states, their resolution of the identity, and the associated Segal-Bargmann transform. I then describe related results including connections to geometric quantization and (1 + 1)-dimensional Yang-Mills theory, the associated coherent states on spheres, and applications to quantum gravity.The third part of this article summarizes recent work of mine with Driver and Kemp on the large-N limit of the Segal-Bargmann transform for the unitary group U (N ). A key result is the identification of the leading-order large-N behavior of the Laplacian on "trace polynomials."

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Heat Kernel Empirical Laws on $${\mathbb {U}}_N$$ U N and $${\mathbb {GL}}_N$$ GL N

Cited by 12 publications

References 37 publications

Stochastic Voronoi Tessellations as Models for Cellular Neighborhoods in Simple Multicellular Organisms

Stochastic Voronoi Tessellations as Models for Cellular Neighborhoods in Simple Multicellular Organisms

A random matrix approach to the Peterson-Thom conjecture

Coherent States for Compact Lie Groups and Their Large-N Limits

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