A probabilistic interpretation of the Macdonald polynomials

@article{Diaconis2010API,
  title={A probabilistic interpretation of the Macdonald polynomials},
  author={Persi Diaconis and Arun Ram},
  journal={Annals of Probability},
  year={2010},
  volume={40},
  pages={1861-1896}
}
The two-parameter Macdonald polynomials are a central object of algebraic combinatorics and representation theory. We give a Markov chain on partitions of k with eigenfunctions the coefficients of the Macdonald polynomials when expanded in the power sum polynomials. The Markov chain has stationary distribution a new two-parameter family of measures on partitions, the inverse of the Macdonald weight (rescaled). The uniform distribution on cycles of permutations and the Ewens sampling formula are… 

Figures and Tables from this paper

Combinatorial Aspects of Macdonald and Related Polynomials
The theory of symmetric functions plays an increasingly important role in modern mathematics, with substantial applications to representation theory, algebraic geometry, special functions,
Markov chains, -trivial monoids and representation theory
TLDR
A general theory of Markov chains realizable as random walks on $\mathscr R$-trivial monoids is developed, and several examples, such as Toom-Tsetlin models, an exchange walk for finite Coxeter groups, as well as examples previously studied by the authors.
From multiline queues to Macdonald polynomials via the exclusion process
Recently James Martin introduced multiline queues, and used them to give a combinatorial formula for the stationary distribution of the multispecies asymmetric simple exclusion exclusion process
Universality of asymptotically Ewens measures on partitions
We give a criterion for functionals of partitions to converge to a universal limit under a class of measures that "behaves like" the Ewens measure. Various limit theorems for the Ewens measure, most
Mixing time of Metropolis chain based on random transposition walk converging to multivariate Ewens distribution.
We prove sharp rates of convergence to the Ewens equilibrium distribution for a family of Metropolis algorithms based on the random transposition shuffle on the symmetric group, with starting point
Hahn polynomials and the Burnside process
We study a natural Markov chain on {0, 1, · · · , n} with eigenvectors the Hahn polynomials. This explicit diagonalization makes it possible to get sharp rates of convergence to stationarity. The
Stationary probabilities of the multispecies TAZRP and modified Macdonald polynomials: I
Recently, a formula for the Macdonald polynomials $P_{\lambda}(X;q,t)$ was given in terms of objects called multiline queues, which also compute probabilities of a particle model from statistical
Measures on Partitions
After brief introduction of the multiplicative measure, defined as a family of measures on integer partitions, which include typical combinatorial structures, this chapter introduces the exponential
A Generating Function Approach to Counting Theorems for Square-Free Polynomials and Maximal Tori
A recent paper of Church, Ellenberg, and Farb uses topology and representation theory of the symmetric group to prove enumerative results about square-free polynomials and F-stable maximal tori of
PROBABILISTIC DIVIDE-AND-CONQUER { A NEW METHOD FOR EXACT SIMULATION { AND LOWER BOUND EXPANSIONS FOR RANDOM BERNOULLI MATRICES VIA NOVEL INTEGER PARTITIONS
This thesis is divided into two areas of combinatorial probability: probabilistic divideand-conquer, and random Bernoulli matrices via novel integer partitions. Probabilistic divide-and-conquer is a
...
1
2
...

References

SHOWING 1-10 OF 87 REFERENCES
Breakthroughs in the theory of Macdonald polynomials.
  • A. Garsia, J. Remmel
  • Mathematics, Medicine
    Proceedings of the National Academy of Sciences of the United States of America
  • 2005
TLDR
The 's, which are now called “Macdonald polynomials,” specialize to many of the well known bases for the symmetric functions, by suitable choices of the parameters q and t.
Combinatorial theory of Macdonald polynomials I: proof of Haglund's formula.
  • J. Haglund, M. Haiman, N. Loehr
  • Mathematics, Medicine
    Proceedings of the National Academy of Sciences of the United States of America
  • 2005
TLDR
A combinatorial proof of this conjecture, which establishes the existence and integrality of H(mu), and obtains the cocharge formula of Lascoux and Schutzenberger for Hall-Littlewood polynomials and formula of Sahi and Knop for Jack's symmetric functions.
A combinatorial formula for Macdonald polynomials
A recursion and a combinatorial formula for Jack polynomials
Heckman and Opdam introduced a non-symmetric analogue of Jack polynomials using Cherednik operators. In this paper, we derive a simple recursion formula for these polynomials and formulas relating
Infinite wedge and random partitions
Abstract. We use representation theory to obtain a number of exact results for random partitions. In particular, we prove a simple determinantal formula for correlation functions of what we call the
A Markov chain on the symmetric group and Jack symmetric functions
  • P. Hanlon
  • Computer Science, Mathematics
    Discret. Math.
  • 1992
A combinatorial formula for Macdonald polynomials
Abstract: We prove a combinatorial formula for the Macdonald polynomial $\tilde{H}_{\mu }(x;q,t)$ which had been conjectured by Haglund. Corollaries to our main theorem include the expansion of
Cherednik algebras, Macdonald polynomials and combinatorics
In the first part of this article we review the general theory of Cherednik algebras and non-symmetric Macdonald polynomials, including a formulation and proof of the fundamental duality theorem in
Dual equivalence graphs, ribbon tableaux and Macdonald polynomials
We make a systematic study of a new combinatorial construction called a dual equivalence graph. Motivated by the dual equivalence relation on standard Young tableaux introduced by Haiman, we
Markov Chains and Mixing Times
This book is an introduction to the modern approach to the theory of Markov chains. The main goal of this approach is to determine the rate of convergence of a Markov chain to the stationary
...
1
2
3
4
5
...