We consider a process of noncolliding
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Abstract q exchangeable random walks on making steps 0 (‘straight’) and −1 (‘down’). A single random walk is called $\mathbb{Z}$q exchangeable if under an elementary transposition of the neighboring steps the probability of the trajectory is multiplied by a parameter $(\text{down},\text{straight})\to (\text{straight},\text{down})$ . Our process of $q\in (0,1)$m noncollidingq exchangeable random walks is obtained from the independentq exchangeable walks via the Doob’sh transform for a nonnegative eigenfunctionh (expressed via theq Vandermonde product) with the eigenvalue less than 1. The system ofm walks evolves in the presence of an absorbing wall at 0. The repulsion mechanism is theq analogue of the Coulomb repulsion of random matrix eigenvalues undergoing Dyson Brownian motion. However, in our model, the particles are confined to the positive halfline and do not spread as Brownian motions or simple random walks. We show that the trajectory of the noncollidingq exchangeable walks started from an arbitrary initial configuration forms a determinantal point process, and express its kernel in a double contour integral form. This kernel is obtained as a limit from the correlation kernel ofq distributed random lozenge tilings of sawtooth polygons. In the limit as , $m\to \mathrm{\infty}$ with $q={e}^{\gamma /m}$γ > 0 fixed, and under a suitable scaling of the initial data, we obtain a limit shape of our noncolliding walks and also show that their local statistics are governed by the incomplete beta kernel. The latter is a distinguished translation invariant ergodic extension of the twodimensional discrete sine kernel. 
We study probability measures on partitions based on symmetric Grothendieck polynomials. These deformations of Schur polynomials introduced in the Ktheory of Grassmannians share many common properties. Our Grothendieck measures are analogs of the Schur measures on partitions introduced by Okounkov (Sel Math 7(1):57–81, 2001). Despite the similarity of determinantal formulas for the probability weights of Schur and Grothendieck measures, we demonstrate that Grothendieck measures are not determinantal point processes. This question is related to the principal minor assignment problem in algebraic geometry, and we employ a determinantal test first obtained by Nanson in 1897 for the 4 × 4 problem. We also propose a procedure for getting Nansonlike determinantal tests for matrices of any size n ≥ 4, which appear new for n ≥ 5. By placing the Grothendieck measures into a new framework of tilted biorthogonal ensembles generalizing a rich class of determinantal processes introduced by Borodin (Nucl Phys B 536:704–732, 1998), we identify Grothendieck random partitions as a crosssection of a Schur process, a determinantal process in two dimensions. This identification expresses the correlation functions of Grothendieck measures through sums of Fredholm determinants, which are not immediately suitable for asymptotic analysis. A more direct approach allows us to obtain a limit shape result for the Grothendieck random partitions. The limit shape curve is not particularly explicit as it arises as a crosssection of the limit shape surface for the Schur process. The gradient of this surface is expressed through the argument of a complex root of a cubic equation.more » « lessFree, publiclyaccessible full text available July 1, 2025

We examine a class of random walks in random environments on Z with bounded jumps, a generalization of the classic onedimensional model. The environments we study have i.i.d. transition probability vectors drawn from Dirichlet distributions. For the transient case of this model, we characterize ballisticity: nonzero limiting velocity. We do this in terms of two parameters, κ0 and κ1. The parameter κ0 governs finite trapping effects. The parameter κ1, which already is known to characterize directional transience, also governs repeated traversals of arbitrarily large regions of the graph. We show that the walk is ballistic if and only if min(κ0, κ1) > 1. We prove some stronger results regarding moments of the quenched Green function and other functions that the quenched Green function dominates. These results help us to better understand the phenomena and parameters affecting ballisticity.more » « lessFree, publiclyaccessible full text available January 1, 2025

We characterize ballistic behavior for general i.i.d. random walks in random envi ronments on Z with bounded jumps. The two characterizations we provide do not use uniform ellipticity conditions. They are natural in the sense that they both relate to formulas for the limiting speed in the nearestneighbor case.more » « less