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Infinite geodesics, competition interfaces and the second class particle in the scaling limit (2025)
Journal Article
Rahman, M., & Virag, B. (2025). Infinite geodesics, competition interfaces and the second class particle in the scaling limit. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 61(2), 1075-1126. https://doi.org/10.1214/23-AIHP1454

We establish fundamental properties of infinite geodesics and competition interfaces in the directed landscape. We construct infinite geodesics in the directed landscape, establish their uniqueness and coalescence, and define Busemann functions. We t... Read More about Infinite geodesics, competition interfaces and the second class particle in the scaling limit.

On inhomogeneous polynuclear growth (2022)
Journal Article
Johansson, K., & Rahman, M. (2022). On inhomogeneous polynuclear growth. Annals of Probability, 50(2), 559-590. https://doi.org/10.1214/21-aop1540

This article studies the inhomogeneous geometric polynuclear growth model, the distribution of which is related to Schur functions. We explain a method to derive its distribution functions in both space-like and time-like directions, focusing on the... Read More about On inhomogeneous polynuclear growth.

Multi-time distribution in discrete polynuclear growth (2021)
Journal Article
Johansson, K., & Rahman, M. (2021). Multi-time distribution in discrete polynuclear growth. Communications on Pure and Applied Mathematics, 74(12), 2561-2627. https://doi.org/10.1002/cpa.21980

We study the multitime distribution in a discrete polynuclear growth model or, equivalently, in directed last‐passage percolation with geometric weights. A formula for the joint multitime distribution function is derived in the discrete setting. It t... Read More about Multi-time distribution in discrete polynuclear growth.

TASEP fluctuations with soft-shock initial data (2020)
Journal Article
Quastel, J., & Rahman, M. (2020). TASEP fluctuations with soft-shock initial data. Annales Henri Lebesgue, 3, 999-1021. https://doi.org/10.5802/ahl.52

We consider the totally asymmetric simple exclusion process with soft-shock initial particle density, which is a step function increasing in the direction of flow and the step size chosen small to admit KPZ scaling. The initial configuration is deter... Read More about TASEP fluctuations with soft-shock initial data.

On the local geometry of graphs in terms of their spectra (2019)
Journal Article
Huang, B., & Rahman, M. (2019). On the local geometry of graphs in terms of their spectra. European Journal of Combinatorics, 81, 378-393. https://doi.org/10.1016/j.ejc.2019.07.001

In this paper we consider the relation between the spectrum and the number of short cycles in large graphs. Suppose G1,G2,G3,... is a sequence of finite and connected graphs that share a common universal cover T and such that the proportion of eigenv... Read More about On the local geometry of graphs in terms of their spectra.

Geometry of Permutation Limits (2019)
Journal Article
Rahman, M., Virág, B., & Vizer, M. (2019). Geometry of Permutation Limits. Combinatorica, 39, 933-960. https://doi.org/10.1007/s00493-019-3817-6

This paper initiates a limit theory of permutation valued processes, building on the recent theory of permutons. We apply this to study the asymptotic behaviour of random sorting networks. We prove that the Archimedean path, the conjectured limit of... Read More about Geometry of Permutation Limits.

Suboptimality of local algorithms for a class of max-cut problems (2019)
Journal Article
Chen, W.-K., Gamarnik, D., Panchenko, D., & Rahman, M. (2019). Suboptimality of local algorithms for a class of max-cut problems. Annals of Probability, 47(3), 1587-1618. https://doi.org/10.1214/18-aop1291

We show that in random K -uniform hypergraphs of constant average degree, for even K ≥ 4 , local algorithms defined as factors of i.i.d. can not find nearly maximal cuts, when the average degree is sufficiently large. These algorithms have been used... Read More about Suboptimality of local algorithms for a class of max-cut problems.

Random sorting networks: local statistics via random matrix laws (2018)
Journal Article
Gorin, V., & Rahman, M. (2019). Random sorting networks: local statistics via random matrix laws. Probability Theory and Related Fields, 175(1-2), 45-96. https://doi.org/10.1007/s00440-018-0886-1

This paper finds the bulk local limit of the swap process of uniformly random sorting networks. The limit object is defined through a deterministic procedure, a local version of the Edelman–Greene algorithm, applied to a two dimensional determinantal... Read More about Random sorting networks: local statistics via random matrix laws.

Local algorithms for independent sets are half-optimal (2017)
Journal Article
Rahman, M., & Virág, B. (2017). Local algorithms for independent sets are half-optimal. Annals of Probability, 45(3), 1543-1577. https://doi.org/10.1214/16-aop1094

We show that the largest density of factor of i.i.d. independent sets in the d -regular tree is asymptotically at most ( log d ) / d as d → ∞ . This matches the lower bound given by previous constructions. It follows that the largest independent sets... Read More about Local algorithms for independent sets are half-optimal.