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Abstract We construct a novel family of difference-permutation operators and prove that they are diagonalized by the wreath MacdonaldP-polynomials; the eigenvalues are written in terms of elementary symmetric polynomials of arbitrary degree. Our operators arise from integral formulas for the action of the horizontal Heisenberg subalgebra in the vertex representation of the corresponding quantum toroidal algebra.
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This content will become publicly available on June 1, 2026
The Large Deviation Principle for W-random spectral measures
The đť‘Š -random graphs provide a flexible framework for modeling large random networks. Using the Large Deviation Principle (LDP) for đť‘Š -random graphs from [19], we prove the LDP for the corresponding class of random symmetric Hilbert-Schmidt integral operators. Our main result describes how the eigenvalues and the eigenspaces of the integral operator are affected by large deviations in the underlying random graphon. To prove the LDP, we demonstrate continuous dependence of the spectral measures associated with integral operators on the corresponding graphons and use the Contraction Principle. To illustrate our results, we obtain leading order asymptotics of the eigenvalues of small-world and bipartite random graphs conditioned on atypical edge counts. These examples suggest several representative scenarios of how the eigenvalues and the eigenspaces are affected by large deviations. We discuss the implications of these observations for bifurcation analysis of Dynamical Systems and Graph Signal Processing.
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- PAR ID:
- 10610590
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Applied and Computational Harmonic Analysis
- Volume:
- 77
- Issue:
- C
- ISSN:
- 1063-5203
- Page Range / eLocation ID:
- 101756
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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