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On the spectral gap of the Kac walk and other binary collision processes
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We give a new and elementary computation of the spectral gap of the Kac walk on the N-sphere. The result is obtained as a by-product of a more general observation which allows to reduce the analysis of the spectral gap of an N-component system to that of the same system for N=3. The method applies to a number of random 'binary collision' processes with complete-graph structure, including non-homogeneous examples such as exclusion and colored exclusion processes with site disorder.
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Joint symmetry and dynamical accessibility in compact Hamiltonian encodings of set cover
A rigorous separation of global, symmetry-allowed, and dynamically accessible spectral gaps for compact Hamiltonian encodings of set cover, including an explicit even-cycle family with an Omega(n^-13) cyclic-gap certificate.
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