{"id":17,"date":"2024-09-24T12:01:30","date_gmt":"2024-09-24T12:01:30","guid":{"rendered":"https:\/\/silicio.math.unifi.it\/wordpress\/eliabisi\/?page_id=17"},"modified":"2026-04-30T14:52:37","modified_gmt":"2026-04-30T12:52:37","slug":"research","status":"publish","type":"page","link":"https:\/\/silicio.math.unifi.it\/wordpress\/eliabisi\/research\/","title":{"rendered":"Research"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\">RESEARCH INTERESTS<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Random matrices<\/li>\n\n\n\n<li>Integrable probability<\/li>\n\n\n\n<li>Stochastic growth models<\/li>\n\n\n\n<li>Interacting particle systems<\/li>\n\n\n\n<li>Last passage percolation<\/li>\n\n\n\n<li>Random polymers<\/li>\n\n\n\n<li>KPZ universality class<\/li>\n\n\n\n<li>Algebraic and enumerative combinatorics<\/li>\n\n\n\n<li>Discrete mathematics<\/li>\n\n\n\n<li>Jacobian conjecture<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\n\n\n\n<h2 class=\"wp-block-heading\">PREPRINTS<\/h2>\n\n\n\n[13] <strong>Semicircle laws with combined variance for non-uniform Erd\u0151s-R\u00e9nyi hypergraphs<\/strong><br>&#8211; with Luca Avena and Eleonora Bordiga.<br>[<a href=\"https:\/\/arxiv.org\/abs\/2604.01877\">arXiv<\/a>]\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p class=\"has-small-font-size\">We consider Erd\u0151s-R\u00e9nyi-type random hypergraphs that are non-uniform, in the sense that hyperedges of different sizes may coexist, and inhomogeneous, in that connection probabilities may depend on the hyperedge size. We study a (collapsed) random adjacency matrix and characterise its limiting spectral distribution under non-sparse conditions as a semicircle law with an explicit parametric variance.<\/p>\n<\/blockquote>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\n\n\n\n<h2 class=\"wp-block-heading\">PEER-REVIEWED PAPERS<\/h2>\n\n\n\n[12]&nbsp;<strong>Random planar trees and the Jacobian conjecture<\/strong><br>&#8211; with Piotr Dyszewski, Nina Gantert, Samuel G. G. Johnston,&nbsp;Joscha Prochno, and&nbsp;Dominik&nbsp;Schmid.<br><em>Journal of the London Mathematical Society<\/em> 113(1), e70416 (2026).<br>[<a href=\"https:\/\/londmathsoc.onlinelibrary.wiley.com\/doi\/10.1112\/jlms.70416\">publication<\/a>] [<a href=\"https:\/\/arxiv.org\/abs\/2301.08221\">arXiv<\/a>]\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p class=\"has-small-font-size\">We develop a probabilistic approach to the celebrated Jacobian conjecture, a more than 80 year old problem from algebraic geometry. We state a stronger conjecture about shuffling subtrees of planar rooted d-ary trees and prove that it is true in a certain asymptotic sense, thereby deducing an approximate version of the Jacobian conjecture.<\/p>\n<\/blockquote>\n\n\n\n[11]&nbsp;<strong>\u03bb-shaped random matrices,&nbsp;\u03bb-plane trees, and&nbsp;\u03bb-Dyck paths<\/strong><br>&#8211; with Fabio D. Cunden.<br><em>Electronic Journal of Probability<\/em> 30, art. no. 11, pp. 1-24 (2025).<br>[<a href=\"https:\/\/doi.org\/10.1214\/25-EJP1268\">publication<\/a>] [<a href=\"https:\/\/arxiv.org\/abs\/2403.07418\">arXiv<\/a>]\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p class=\"has-small-font-size\">For a given Young diagram \u03bb, a \u03bb-shaped random matrix is a random matrix whose entries are i.i.d. in the boxes of \u03bb and zero outside. We study the limiting spectral distribution of \u03bb-shaped random matrices, where \u03bb is a self-conjugate Young diagram. The moments of this distribution enumerate two combinatorial objects:&nbsp;\u03bb-plane trees and&nbsp;\u03bb-Dyck paths.<\/p>\n<\/blockquote>\n\n\n\n[10]&nbsp;<strong>Matsumoto-Yor and Dufresne type theorems for a random walk on positive definite matrices<\/strong><br>&#8211; with&nbsp;Jonas Arista and Neil O&#8217;Connell.<br><em>Annales de l&#8217;Institut Henri Poincar\u00e9, Probabilit\u00e9s et Statistiques<\/em> 60(2),&nbsp;pp. 923-945&nbsp;(2024).<br>[<a href=\"https:\/\/doi.org\/10.1214\/22-AIHP1338\">publication<\/a>] [<a href=\"https:\/\/arxiv.org\/abs\/2112.12558\">arXiv<\/a>]\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p class=\"has-small-font-size\">We establish analogues of the&nbsp;Matsumoto-Yor&nbsp;theorem and of the&nbsp;Dufresne identity (for integrals of exponential functional of Brownian motion)&nbsp;in the context of a multiplicative&nbsp;random walk on positive definite matrices.<\/p>\n<\/blockquote>\n\n\n\n[9]&nbsp;<strong>Matrix Whittaker processes<\/strong><br>&#8211;&nbsp;with Jonas Arista and Neil O&#8217;Connell.<br><em>Probability Theory and Related Fields<\/em> 187, pp. 203-257 (2023).<br>[<a href=\"https:\/\/link.springer.com\/article\/10.1007\/s00440-023-01210-y\">publication<\/a>] [<a href=\"https:\/\/arxiv.org\/abs\/2203.14868\">arXiv<\/a>]<br><mark style=\"background-color:#ffffff\" class=\"has-inline-color has-vivid-red-color\">TU Wien 2023 <strong>Best Paper Award<\/strong> in Mathematics<\/mark><\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p class=\"has-small-font-size\">We study certain&nbsp;matrix-valued interacting random walks, which generalise particle systems with pushing and blocking mechanisms and can also be viewed as high-dimensional analogues of&nbsp;log-gamma polymer&nbsp;partition functions. We show that marginals of these processes exhibit&nbsp;certain &#8216;matrix Whittaker measures&#8217;.<\/p>\n<\/blockquote>\n\n\n\n[8]<strong>&nbsp;Non-intersecting path constructions for TASEP with inhomogeneous rates and the KPZ fixed point<\/strong><br>&#8211; with Yuchen Liao, Axel Saenz, and Nikos Zygouras.<br><em>Communications in Mathematical Physics<\/em> 402, pp. 285-333 (2023).<br>[<a href=\"https:\/\/link.springer.com\/article\/10.1007\/s00220-023-04723-8\">publication<\/a>] [<a href=\"https:\/\/arxiv.org\/abs\/2208.13580\">arXiv<\/a>]\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p class=\"has-small-font-size\">We consider a discrete-time Totally Asymmetric Simple Exclusion Process with particle-dependent and time-inhomogeneous jump rates. Using the RSK algorithm, intertwining relations and non-intersecting path constructions, we obtain a representation for the correlation kernel of the particle positions in terms of random walk hitting probabilities, \u00e0 la&nbsp;<a href=\"https:\/\/www.intlpress.com\/site\/pub\/pages\/journals\/items\/acta\/content\/vols\/0227\/0001\/a003\/index.php\">Matetski-Quastel-Remenik<\/a>.<\/p>\n<\/blockquote>\n\n\n\n[7]<strong>&nbsp;Transition between characters of classical groups, decomposition of Gelfand-Tsetlin patterns and last passage percolation<\/strong><br>&#8211; with&nbsp;Nikos Zygouras.<br><em>Advances in Mathematics<\/em> 404, Part B,&nbsp;108453 (2022).<br>[<a href=\"https:\/\/doi.org\/10.1016\/j.aim.2022.108453\">publication<\/a>] [<a href=\"https:\/\/arxiv.org\/abs\/1905.09756\">arXiv<\/a>]\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p class=\"has-small-font-size\">We establish numerous formulas for last passage percolation models with various symmetries in terms of irreducible characters of the classical groups and, more generally, in terms of certain interpolating symmetric polynomials.&nbsp;At the level of KPZ asymptotic analysis, these formulas yield new routes to universal random matrix limiting distributions, providing&nbsp;a structural explanation of the duality between their Pfaffian and determinant formulations.<\/p>\n<\/blockquote>\n\n\n\n[6]<strong>&nbsp;The oriented swap process and last passage percolation<\/strong><br>&#8211; with Fabio D. Cunden, Shane Gibbons, and Dan Romik.<br><em>Random Structures and Algorithms<\/em> 60(4), pp.&nbsp;690-715&nbsp;(2022).<br>[<a href=\"https:\/\/doi.org\/10.1002\/rsa.21055\">publication<\/a>] [<a href=\"https:\/\/arxiv.org\/abs\/2005.02043\">arXiv<\/a>]\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p class=\"has-small-font-size\">We study some connections between the&nbsp;oriented swap process, the&nbsp;corner growth model, and the&nbsp;last passage percolation&nbsp;model.&nbsp;We also conjecture a distributional identity between these models, or equivalently a purely combinatorial identity which involves&nbsp;sorting networks&nbsp;and&nbsp;staircase shape Young tableaux&nbsp;and is related to the celebrated&nbsp;Edelman-Greene bijection.<\/p>\n<\/blockquote>\n\n\n\n[5]<strong>&nbsp;The geometric Burge correspondence and the partition function of polymer replicas<\/strong><br>&#8211; with Neil O&#8217;Connell and Nikos Zygouras.<br><em>Selecta Mathematica New Series<\/em> 27, art. no. 100 (2021).<br>[<a href=\"https:\/\/doi.org\/10.1007\/s00029-021-00712-8\">publication<\/a>] [<a href=\"https:\/\/arxiv.org\/abs\/2001.09145\">arXiv<\/a>]\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p class=\"has-small-font-size\">We construct a&nbsp;geometric lifting&nbsp;of the&nbsp;Burge correspondence&nbsp;(a version of the RSK algorithm) as a composition of local birational maps. We apply our construction to a model of two polymer paths of given length constrained to have the same endpoint, known as&nbsp;polymer replica, proving&nbsp;that the distribution of the polymer replica partition function in a log-gamma random environment is &nbsp;Whittaker measure.<\/p>\n<\/blockquote>\n\n\n\n[4]&nbsp;<strong>Sorting networks, staircase Young tableaux and last passage percolation<\/strong><br>&#8211; with&nbsp;Fabio D. Cunden, Shane Gibbons, and Dan Romik.<br><em>S\u00e9minaire Lotharingien de Combinatoire 84B, Proceedings of the 32nd Conference on Formal Power Series and Algebraic Combinatorics<\/em>, art. no. 3 (2020).<br>[<a href=\"https:\/\/www.mat.univie.ac.at\/~slc\/wpapers\/FPSAC2020\/3.html\">publication<\/a>]<em>&nbsp;<\/em>[<a href=\"https:\/\/arxiv.org\/abs\/2003.03331\">arXiv<\/a>]\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p class=\"has-small-font-size\">Extended abstract of [6].<\/p>\n<\/blockquote>\n\n\n\n[3]&nbsp;<strong>GOE and Airy<sub>2\u21921<\/sub>&nbsp;marginal distribution via symplectic Schur functions<\/strong><br>&#8211; with&nbsp;Nikos Zygouras.<br><em>Probability and Analysis in Interacting Physical Systems &#8211;&nbsp;In Honor of S.R.S. Varadhan<\/em><em>&nbsp;<\/em><em>(e<\/em><em>ditors: P. Friz, W.&nbsp;K\u00f6nig, C.&nbsp;Mukherjee, S.&nbsp;Olla),&nbsp;Springer<\/em>,&nbsp;pp. 191-213&nbsp;(2019).<br>[<a href=\"http:\/\/dx.doi.org\/10.1007\/978-3-030-15338-0_7\">publication<\/a>]&nbsp;[<a href=\"https:\/\/arxiv.org\/abs\/1711.05120\">arXiv<\/a>]\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p class=\"has-small-font-size\">Building on the symplectic Schur formulas found in [2], we perform a scaling limit of the point-to-line and point-to-half-line last passage percolation models with exponential weights, obtaining the GOE e Airy<sub>2\u21921&nbsp;<\/sub>distributions.<\/p>\n<\/blockquote>\n\n\n\n[2]<strong>&nbsp;Point-to-line polymers and orthogonal Whittaker functions<\/strong><br>&#8211; with Nikos Zygouras.<br><em>Transactions of the American Mathematical Society<\/em> 371(12), pp. 8339-8379 (2019).<br>[<a href=\"https:\/\/doi.org\/10.1090\/tran\/7423\">publication<\/a>] [<a href=\"https:\/\/arxiv.org\/abs\/1703.07337\">arXiv<\/a>]\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p class=\"has-small-font-size\">We study some KPZ integrable models on the 2D integer lattice (log-gamma polymer&nbsp;and exponential&nbsp;last passage percolation) with various point-to-line path geometries. We find novel connections with special functions from algebraic combinatorics and representation theory, i.e.&nbsp;orthogonal Whittaker functions&nbsp;and&nbsp;symplectic Schur functions.<\/p>\n<\/blockquote>\n\n\n\n[1]&nbsp;<strong>Symbolic analysis of higher-order side channel countermeasures<\/strong><br>&#8211; with Filippo Melzani and Vittorio Zaccaria.<br><em>IEEE Transactions on Computers<\/em> 66(6), pp. 1099-1105 (2017).<br>[<a href=\"https:\/\/doi.org\/10.1109\/TC.2016.2635650\">publication<\/a>]\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p class=\"has-small-font-size\">Side channel attacks exploit vulnerable leakage combinations, i.e. sets of data physically leaked during the execution of a cryptographic implementation (for example in the form of power consumption),&nbsp;that statistically depend on sensitive data and involve a part of the secret key. This work&nbsp;presents equivalent and easy-to-verify conditions for the&nbsp;vulnerability of leakage combinations, useful to validate the protection order guaranteed by an implementation.<\/p>\n<\/blockquote>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\n\n\n\n<h2 class=\"wp-block-heading\">REPORTS<\/h2>\n\n\n\n<p><strong>Random matrices, Young diagrams, and trees<\/strong><br><em>Mini-Workshop: Mixing Times in the Kardar-Parisi-Zhang Universality<\/em> <em>Class<\/em>.<br><em>Oberwolfach Rep.<\/em> 51&nbsp;(2024).<br>[<a href=\"https:\/\/publications.mfo.de\/handle\/mfo\/4267\">publication<\/a>]\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p class=\"has-small-font-size\">Extended abstract of [11].<\/p>\n<\/blockquote>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\n\n\n\n<h2 class=\"wp-block-heading\">PHD THESIS<\/h2>\n\n\n\n<p><strong>Random polymers via orthogonal Whittaker and symplectic Schur functions<\/strong><br>Submitted and defended in 2018.<br>[<a href=\"http:\/\/wrap.warwick.ac.uk\/121448\/\">Warwick repository<\/a>] [<a href=\"https:\/\/arxiv.org\/abs\/1810.03734\">arXiv<\/a>]\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p class=\"has-small-font-size\">My PhD thesis is concerned with integrable polymer and last passage percolation models with point-to-line path geometries and their connections with orthogonal Whittaker and symplectic Schur functions. It is an extended versions of the articles [2] and [3]; in addition, it studies the analogous last passage percolation models with geometric weights.<\/p>\n<\/blockquote>\n","protected":false},"excerpt":{"rendered":"<p>RESEARCH INTERESTS PREPRINTS [13] Semicircle laws with combined variance for non-uniform Erd\u0151s-R\u00e9nyi hypergraphs&#8211; with Luca Avena and Eleonora Bordiga.[arXiv] We consider Erd\u0151s-R\u00e9nyi-type random hypergraphs that are non-uniform, in the sense that hyperedges of different sizes may coexist, and inhomogeneous, in that connection probabilities may depend on the hyperedge size. We study a (collapsed) random adjacency &hellip; <a href=\"https:\/\/silicio.math.unifi.it\/wordpress\/eliabisi\/research\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Research<\/span><\/a><\/p>\n","protected":false},"author":17,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-17","page","type-page","status-publish","hentry","without-featured-image"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v23.4 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Research - Elia Bisi<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/silicio.math.unifi.it\/wordpress\/eliabisi\/research\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Research - Elia Bisi\" \/>\n<meta property=\"og:description\" content=\"RESEARCH INTERESTS PREPRINTS [13] Semicircle laws with combined variance for non-uniform Erd\u0151s-R\u00e9nyi hypergraphs&#8211; with Luca Avena and Eleonora Bordiga.[arXiv] We consider Erd\u0151s-R\u00e9nyi-type random hypergraphs that are non-uniform, in the sense that hyperedges of different sizes may coexist, and inhomogeneous, in that connection probabilities may depend on the hyperedge size. 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We study a (collapsed) random adjacency &hellip; Continue reading Research","og_url":"https:\/\/silicio.math.unifi.it\/wordpress\/eliabisi\/research\/","og_site_name":"Elia Bisi","article_modified_time":"2026-04-30T12:52:37+00:00","twitter_card":"summary_large_image","twitter_misc":{"Est. reading time":"2 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/silicio.math.unifi.it\/wordpress\/eliabisi\/research\/","url":"https:\/\/silicio.math.unifi.it\/wordpress\/eliabisi\/research\/","name":"Research - Elia Bisi","isPartOf":{"@id":"https:\/\/silicio.math.unifi.it\/wordpress\/eliabisi\/#website"},"datePublished":"2024-09-24T12:01:30+00:00","dateModified":"2026-04-30T12:52:37+00:00","breadcrumb":{"@id":"https:\/\/silicio.math.unifi.it\/wordpress\/eliabisi\/research\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/silicio.math.unifi.it\/wordpress\/eliabisi\/research\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/silicio.math.unifi.it\/wordpress\/eliabisi\/research\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/silicio.math.unifi.it\/wordpress\/eliabisi\/"},{"@type":"ListItem","position":2,"name":"Research"}]},{"@type":"WebSite","@id":"https:\/\/silicio.math.unifi.it\/wordpress\/eliabisi\/#website","url":"https:\/\/silicio.math.unifi.it\/wordpress\/eliabisi\/","name":"Elia Bisi","description":"Mathematician","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/silicio.math.unifi.it\/wordpress\/eliabisi\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"}]}},"_links":{"self":[{"href":"https:\/\/silicio.math.unifi.it\/wordpress\/eliabisi\/wp-json\/wp\/v2\/pages\/17"}],"collection":[{"href":"https:\/\/silicio.math.unifi.it\/wordpress\/eliabisi\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/silicio.math.unifi.it\/wordpress\/eliabisi\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/silicio.math.unifi.it\/wordpress\/eliabisi\/wp-json\/wp\/v2\/users\/17"}],"replies":[{"embeddable":true,"href":"https:\/\/silicio.math.unifi.it\/wordpress\/eliabisi\/wp-json\/wp\/v2\/comments?post=17"}],"version-history":[{"count":22,"href":"https:\/\/silicio.math.unifi.it\/wordpress\/eliabisi\/wp-json\/wp\/v2\/pages\/17\/revisions"}],"predecessor-version":[{"id":299,"href":"https:\/\/silicio.math.unifi.it\/wordpress\/eliabisi\/wp-json\/wp\/v2\/pages\/17\/revisions\/299"}],"wp:attachment":[{"href":"https:\/\/silicio.math.unifi.it\/wordpress\/eliabisi\/wp-json\/wp\/v2\/media?parent=17"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}