A Randomized Approximation Algorithm for Doubly Free-Boundary Macdonald Processes
Abstract
We construct an approximate sampler for finite doubly free-boundary Macdonald rail-yard processes with Macdonald, boundary, and edge parameters in (0,1). The construction allows arbitrary finite sequences of additions and removals of horizontal or vertical strips and all four boundary types. It reverses a finite commutation--reflection history from a partition sequence whose entries are equal, with the common partition weighted by both boundaries. We derive local reconstruction kernels from the skew Cauchy, Littlewood, and branching identities. Bulk and boundary updates combine integer increments with conditional shape sampling at a prescribed size; same-sign swaps use finite conditional resampling. For K reflection periods and an initial partition-size cutoff M, the algorithm terminates almost surely and samples the resulting approximating law exactly. We prove total variation convergence to the target process as K,M→∞ and give explicit bounds for both truncation errors, decaying geometrically in the cutoffs for fixed input parameters. Under fixed parameter bounds, we also bound the expected operation count of direct enumeration in a unit-cost model. We further construct an exact rejection sampler for the initial partition law without a size cutoff or evaluation of its normalizing constant. Using this sampler removes the size-truncation error, leaving only the error from the finite reflection history.
Related
- Gaussian approximation for the integrated intensity of counting processes: application to Hawkes processes
- An energy-decreasing algorithm for the finite element approximation of ferronematic equilibrium states and its numerical realization
- Periodic measures for time-inhomogeneous stochastic differential equations driven by multiplicative α-stable processes: Existence, uniqueness and numerical approximation
- A tight 1/3-approximation algorithm and fully polynomial-time approximation schemes for the Colored Knapsack Problem
- Approximation algorithm for mixed volumes of a constant number of convex bodies
BibTeX
@misc{li2026randomized,
title = {{A Randomized Approximation Algorithm for Doubly Free-Boundary Macdonald Processes}},
author = {Zhongyang Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18698},
eprint = {2610.18698},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18698},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Randomized Approximation Algorithm for Doubly Free-Boundary Macdonald Processes
AU - Zhongyang Li
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18698
UR - https://arxiv.org/abs/2610.18698
AB - We construct an approximate sampler for finite doubly free-boundary Macdonald rail-yard processes with Macdonald, boundary, and edge parameters in (0,1). The construction allows arbitrary finite sequences of additions and removals of horizontal or vertical strips and all four boundary types. It reverses a finite commutation--reflection history from a partition sequence whose entries are equal, with the common partition weighted by both boundaries. We derive local reconstruction kernels from the skew Cauchy, Littlewood, and branching identities. Bulk and boundary updates combine integer increments with conditional shape sampling at a prescribed size; same-sign swaps use finite conditional resampling. For K reflection periods and an initial partition-size cutoff M, the algorithm terminates almost surely and samples the resulting approximating law exactly. We prove total variation convergence to the target process as K,M→∞ and give explicit bounds for both truncation errors, decaying geometrically in the cutoffs for fixed input parameters. Under fixed parameter bounds, we also bound the expected operation count of direct enumeration in a unit-cost model. We further construct an exact rejection sampler for the initial partition law without a size cutoff or evaluation of its normalizing constant. Using this sampler removes the size-truncation error, leaving only the error from the finite reflection history.
ER -