[
    {
        "id": "authors:xen46-etw49",
        "collection": "authors",
        "collection_id": "xen46-etw49",
        "cite_using_url": "https://authors.library.caltech.edu/records/xen46-etw49",
        "type": "article",
        "title": "A convergence framework for Airy\u03b2 line ensemble via pole evolution",
        "author": [
            {
                "family_name": "Huang",
                "given_name": "Jiaoyang",
                "orcid": "0000-0001-8266-1301"
            },
            {
                "family_name": "Zhang",
                "given_name": "Lingfu",
                "orcid": "0000-0002-4794-7678",
                "clpid": "Zhang-Lingfu"
            }
        ],
        "abstract": "<p>The Airy &beta; line ensemble is an infinite sequence of random curves. It is a natural extension of the Tracy-Widom &beta; distributions, and is expected to be the universal edge scaling limit of a range of models in random matrix theory and statistical mechanics. In this work, we provide a framework of proving convergence to the Airy &beta; line ensemble, via a characterization through the pole evolution of meromorphic functions satisfying certain stochastic differential equations. Our framework is then applied to prove the universality of the Airy &beta; line ensemble as the edge limit of various continuous time processes, including Dyson Brownian motions with general &beta; and potentials, Laguerre processes and Jacobi processes.</p>",
        "doi": "10.1016/j.aim.2026.111028",
        "issn": "0001-8708",
        "publisher": "Elsevier",
        "publication": "Advances in Mathematics",
        "publication_date": "2026-07",
        "volume": "498",
        "pages": "111028"
    },
    {
        "id": "authors:wak0c-vbx30",
        "collection": "authors",
        "collection_id": "wak0c-vbx30",
        "cite_using_url": "https://authors.library.caltech.edu/records/wak0c-vbx30",
        "type": "article",
        "title": "Convergence of the environment seen from geodesics in exponential last-passage percolation",
        "author": [
            {
                "family_name": "Martin",
                "given_name": "James B."
            },
            {
                "family_name": "Sly",
                "given_name": "Allan"
            },
            {
                "family_name": "Zhang",
                "given_name": "Lingfu",
                "orcid": "0000-0002-4794-7678",
                "clpid": "Zhang-Lingfu"
            }
        ],
        "abstract": "A well-known question in planar first-passage percolation concerns the convergence of the empirical distribution of weights as seen along geodesics. We demonstrate this convergence for an explicit model, directed last-passage percolation on \n            \n              \\mathbb{Z}^{2}\n            \n             with i.i.d. exponential weights, and provide explicit formulae for the limiting distributions, which depend on the asymptotic direction. For example, for geodesics in the direction of the diagonal, the limiting weight distribution has density \n            \n              (1/4+x/2+x^{2}/8)e^{-x}\n            \n            , and so is a mixture of Gamma(\n            \n              1,1\n            \n            ), Gamma(\n            \n              2,1\n            \n            ), and Gamma(\n            \n              3,1\n            \n            ) distributions with weights \n            \n              1/4\n            \n            , \n            \n              1/2\n            \n            , and \n            \n              1/4\n            \n             respectively. More generally, we study the local environment as seen from vertices along geodesics (including information about the shape of the path and about the weights on and off the path in a local neighborhood). We consider finite geodesics from \n            \n              (0,0)\n            \n             to \n            \n              n\\boldsymbol{\\rho}\n            \n             for some vector \n            \n              \\boldsymbol{\\rho}\n            \n             in the first quadrant, in the limit as \n            \n              n\\to\\infty\n            \n            , as well as semi-infinite geodesics in direction \n            \n              \\boldsymbol{\\rho}\n            \n            . We show almost sure convergence of the empirical distributions of the environments along these geodesics, as well as convergence of the distributions of the environment around a typical point in these geodesics, to the same limiting distribution, for which we give an explicit description.We make extensive use of a correspondence with TASEP as seen from an isolated second-class particle for which we prove new results concerning ergodicity and convergence to equilibrium. Our analysis relies on geometric arguments involving estimates for last-passage times, available from the integrable probability literature.",
        "doi": "10.4171/jems/1594",
        "issn": "1435-9855",
        "publisher": "European Mathematical Society - EMS - Publishing House GmbH",
        "publication": "Journal of the European Mathematical Society",
        "publication_date": "2025-03-06",
        "series_number": "3",
        "volume": "27",
        "issue": "3",
        "pages": "877\u2013970"
    }
]