Macdonald processes

By Alexei Borodin, Ivan Corwin. Probability theory and related fields, 2011

438 citations recorded by the source corpus.

Macdonald processes are probability measures on sequences of partitions defined in terms of nonnegative specializations of the Macdonald symmetric functions and two Macdonald parameters $$q,t \in [0,1)$$. We prove several results about these processes, which include the following. (1) We explicitly evaluate expectations of a rich family of observables for these processes. (2) In the case $$t=0$$, we find a Fredholm determinant formula for a $$q$$-Laplace transform of the distribution of the last part of the Macdonald-random partition. (3) We introduce Markov dynamics that preserve the class of Macdonald processes and lead to new “integrable” 2d and 1d interacting particle systems. (4) In a large time limit transition, and as $$q$$ goes to 1, the particles of these systems crystallize on a lattice, and fluctuations around the lattice converge to O’Connell’s Whittaker process that describe semi-discrete Brownian directed polymers. (5) This yields a Fredholm determinant for the Laplace transform of the polymer partition function, and taking its asymptotics we prove KPZ universality for the polymer (free energy fluctuation exponent $$1/3$$ and Tracy-Widom GUE limit law). (6) Under intermediate disorder scaling, we recover the Laplace transform of the solution of the KPZ equation with narrow wedge initial data. (7) We provide contour integral formulas for a wide array of polymer moments. (8) This results in a new ansatz for solving quantum many body systems such as the delta Bose gas.

Subfields: Artificial Intelligence, Discrete Mathematics and Combinatorics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/e8dcb9dc22f03cb0001005b3c767bf5485c2f209

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. Jeremy Quastel
2. Ivan Corwin
3. P. Ferrari
4. Alexei Borodin
5. B. Vető
6. L. Petrov
7. Robert C. Dalang
8. Le Chen

In Discrete Mathematics and Combinatorics

1. L. Petrov
2. Ivan Corwin
3. P. Ferrari
4. Jeremy Quastel
5. Robert C. Dalang
6. Le Chen
7. Alexei Borodin
8. B. Vető

In Statistics and Probability

1. Ivan Corwin
2. Jeremy Quastel
3. Alexei Borodin
4. P. Ferrari
5. L. Petrov
6. Herbert Spohn
7. B. Vető
8. K. Takeuchi

Citations in the corpus

• Free Energy Fluctuations for Directed Polymers in Random Media in 1 + 1 Dimension (2012, methodology): “The second contribution of this work is to develop a variant on the formulas of [15] for the semi-discrete polymer which are well-adapted for rigorous asymptotic analysis at all temperatures (and for boundary perturbations).”
• Introduction to KPZ (2011, background): “A very recent article [COSZ11] discusses many such exact formulas in the context of the tropical version of RSK (see also [BC11] for very general results about solvability in this context.”
• Height Fluctuations for the Stationary KPZ Equation (2014, methodology): “This is made by analyzing a semidiscrete directed polymer model, which in its turn is obtained as a limit of the q-Whittaker process [6] with appropriate initial measure.”
• Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions (2013, methodology): “See [2, 3, 4, 5, 8, 9, 10, 17, 20] for the intermittency problem, [15, 16] for probabilistic potential theory, [31, 32] for regularity of the solution, and [26, 27, 29, 33] for several other properties.”
• Stochastic Higher Spin Vertex Models on the Line (2015, methodology): “when τ < 1 (in other words, case (2) above where q ∈ (1,∞)) and when ~g(t) is started from the step initial data (gi(0) = 1i≥1). Building on a combination of the approaches developed in [TW08] and [BC14], they compute a contour integral formula for E[τLN↓n(~g)], L ∈ Z ≥1, and eventually utilize t”
• The Ubiquitous Ewens Sampling Formula (2016, background): “Macdonald processes [13] are a particularly interesting family of probability distributions on sequences of integer partitions that arise in certain models for interacting particle systems and directed random polymers in statistical physics.”
• Bernstein-gamma functions and exponential functionals of Levy Processes (2016, background): “Diaconis and Fill [23] in relation with strong uniform times, by Carmona, Petit and Yor [21] in relation to the so-called selfsimilar saw tooth-processes, by Borodin and Corwin [19] in the context of Macdonald processes, by Pal and Shkolnikov [50] for linking diffusion operators, and, by Patie and S”
• The One-Dimensional KPZ Equation and Its Universality Class (2015, background): “Similar formulas appear in [67] as solutions to (4.”