Research

RESEARCH INTERESTS

  • Integrable probability
  • Stochastic growth models
  • Interacting particle systems
  • Last passage percolation
  • Random polymers
  • KPZ universality class
  • Random matrices
  • Algebraic and enumerative combinatorics
  • Discrete mathematics
  • Representation theory
  • Jacobian conjecture

PREPRINTS

[12] λ-shaped random matrices, λ-plane trees, and λ-Dyck paths
– with Fabio D. Cunden.
Submitted (2024).
[arXiv]

For a given Young diagram λ, a λ-shaped random matrix is a random matrix whose entries are i.i.d. in the boxes of λ and zero outside. We study the limiting spectral distribution of λ-shaped random matrices, where λ is a self-conjugate Young diagram. The moments of this distribution enumerate two combinatorial objects: λ-plane trees and λ-Dyck paths.

[11] Random planar trees and the Jacobian conjecture
– with Piotr Dyszewski, Nina Gantert, Samuel G. G. Johnston, Joscha Prochno, and Dominik Schmid.
Submitted (2023).
[arXiv]

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.

PUBLISHED/ACCEPTED PAPERS

[10] Matrix Whittaker processes
– with Jonas Arista and Neil O’Connell.
Probability Theory and Related Fields, vol. 187, pp. 203-257 (2023).
[publication] [arXiv]

We study certain matrix-valued interacting random walks, which generalise particle systems with pushing and blocking mechanisms and can also be viewed as high-dimensional analogues of log-gamma polymer partition functions. We show that marginals of these processes exhibit certain ‘matrix Whittaker measures’.

[9] Non-intersecting path constructions for TASEP with inhomogeneous rates and the KPZ fixed point
– with Yuchen Liao, Axel Saenz, and Nikos Zygouras.
Communications in Mathematical Physics, vol. 402, pp. 285-333 (2023).
[publication] [arXiv]

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, à la Matetski-Quastel-Remenik.

[8] Matsumoto-Yor and Dufresne type theorems for a random walk on positive definite matrices
– with Jonas Arista and Neil O’Connell.
Annales de l’Institut Henri Poincaré, Probabilités et Statistiques (2022+).
[arXiv]

We establish analogues of the Matsumoto-Yor theorem and of the Dufresne identity (for integrals of exponential functional of Brownian motion) in the context of a multiplicative random walk on positive definite matrices.

[7] Transition between characters of classical groups, decomposition of Gelfand-Tsetlin patterns and last passage percolation
– with Nikos Zygouras.
Advances in Mathematics, vol. 404, Part B, 108453 (2022).
[publication] [arXiv]

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. At the level of KPZ asymptotic analysis, these formulas yield new routes to universal random matrix limiting distributions, providing a structural explanation of the duality between their Pfaffian and determinant formulations.

[6] The oriented swap process and last passage percolation
– with Fabio D. Cunden, Shane Gibbons, and Dan Romik.
Random Structures and Algorithms, vol. 60, no. 4, pp. 690-715 (2022).
[publication] [arXiv]

We study some connections between the oriented swap process, the corner growth model, and the last passage percolation model. We also conjecture a distributional identity between these models, or equivalently a purely combinatorial identity which involves sorting networks and staircase shape Young tableaux and is related to the celebrated Edelman-Greene bijection.

[5] The geometric Burge correspondence and the partition function of polymer replicas
– with Neil O’Connell and Nikos Zygouras.
Selecta Mathematica New Series, vol. 27, art. #100 (2021).
[publication] [arXiv]

We construct a geometric lifting of the Burge correspondence (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 polymer replica, proving that the distribution of the polymer replica partition function in a log-gamma random environment is  Whittaker measure.

[4] Sorting networks, staircase Young tableaux and last passage percolation
– with Fabio D. Cunden, Shane Gibbons, and Dan Romik.
Séminaire Lotharingien de Combinatoire 84B, Proceedings of the 32nd Conference on Formal Power Series and Algebraic Combinatorics, art. #3 (2020).
[publication] [arXiv]

Extended abstract of [6].

[3] GOE and Airy2→1 marginal distribution via symplectic Schur functions
– with Nikos Zygouras.
Probability and Analysis in Interacting Physical Systems – In Honor of S.R.S. Varadhan (editors: P. Friz, W. König, C. Mukherjee, S. Olla), Springer, pp. 191-213 (2019).
[publication] [arXiv]

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 Airy2→1 distributions.

[2] Point-to-line polymers and orthogonal Whittaker functions
– with Nikos Zygouras.
Transactions of the American Mathematical Society, vol. 371, no. 12, pp. 8339-8379 (2019).
[publication] [arXiv]

We study some KPZ integrable models on the 2D integer lattice (log-gamma polymer and exponential 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. orthogonal Whittaker functions and symplectic Schur functions.

[1] Symbolic analysis of higher-order side channel countermeasures
– with Filippo Melzani and Vittorio Zaccaria.
IEEE Transactions on Computers, vol. 66, no. 6, pp. 1099-1105 (2017).
[publication]

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), that statistically depend on sensitive data and involve a part of the secret key. This work presents equivalent and easy-to-verify conditions for the vulnerability of leakage combinations, useful to validate the protection order guaranteed by an implementation.

PHD THESIS

Random polymers via orthogonal Whittaker and symplectic Schur functions
Submitted and defended in 2018.
[Warwick repository] [arXiv]

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.