Random Shuffles and Group Representations

Flatto, Leopold; Odlyzko, Andrew; Wales, David B.

doi:10.1214/aop/1176993073

Cited by 87 publications

(100 citation statements)

References 10 publications

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“…These cases of the Aldous-Diaconis conjecture were previously know [2,8], as well as some other examples where one can compute λ(n) exactly [3,5,4]. Needless to say, a full proof of FOEL, the Aldous-Diaconis Conjecture, or even a proof for additionial special cases, would be of great interest.…”

Section: Conjecturementioning

confidence: 73%

Ordering of Energy Levels in Heisenberg Models and Applications

Nachtergaele

Starr

2006

Mathematical Physics of Quantum Mechanics

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Summary. In a recent paper [17] we conjectured that for ferromagnetic Heisenberg models the smallest eigenvalues in the invariant subspaces of fixed total spin are monotone decreasing as a function of the total spin and called this property ferromagnetic ordering of energy levels (FOEL). We have proved this conjecture for the Heisenberg model with arbitrary spins and coupling constants on a chain [17,20]. In this paper we give a pedagogical introduction to this result and also discuss some extensions and implications. The latter include the property that the relaxation time of symmetric simple exclusion processes on a graph for which FOEL can be proved, equals the relaxation time of a random walk on the same graph. This equality of relaxation times is known as Aldous' Conjecture.

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Section: Conjecturementioning

confidence: 73%

Ordering of Energy Levels in Heisenberg Models and Applications

Nachtergaele

Starr

2006

Mathematical Physics of Quantum Mechanics

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“…The method also works for less symmetric problems: Flatto et al (14) showed a cutoff at n(log n + 6) for "transpose random with top." Lulov (15) studied the following problem.…”

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confidence: 99%

The cutoff phenomenon in finite Markov chains.

Diaconis

1996

Proc. Natl. Acad. Sci. U.S.A.

293

261

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Natural mixing processes modeled by Markov chains often show a sharp cutoff in their convergence to long-time behavior. This paper presents problems where the cutoff can be proved (card shuffling, the Ehrenfests' urn). It shows that chains with polynomial growth (drunkard's walk) do not show cutoffs. The best general understanding of such cutoffs (high multiplicity of second eigenvalues due to symmetry) is explored. Examples are given where the symmetry is broken but the cutoff phenomenon persists.

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“…For various applications of this rule cf. [FL,FOW,JK2.7,Ro1,St7.4,and SW]. The classical proofs of this rule use the Littlewood Richardson rule [J,Chap.…”

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confidence: 99%

A Recursive Rule for Kazhdan–Lusztig Characters

Roichman

1997

Advances in Mathematics

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Random Shuffles and Group Representations

Cited by 87 publications

References 10 publications

Ordering of Energy Levels in Heisenberg Models and Applications

Ordering of Energy Levels in Heisenberg Models and Applications

The cutoff phenomenon in finite Markov chains.

A Recursive Rule for Kazhdan–Lusztig Characters

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