Integrable probability

My research interests lie in probability theory, mainly in discrete probabilistic models that arise from statistical physics, e.g. random polymers, stochastic growth models, and interacting particle systems. I am especially interested in those models that disclose fertile connections with other areas of mathematics such as algebraic combinatorics, random matrices, and representation theory.

I became particularly interested in the Kardar-Parisi-Zhang (KPZ) universality class, a large family of probabilistic models that share macroscopic features such as the fluctuations exponent 1/3 and limiting distributions from random matrix theory. I studied two KPZ models in particular: last passage percolation and random directed polymers, which can be seen as probability measures on directed lattice paths in a random environment. The extremely rich mathematical structure behind the integrable versions of these models has also lead me to independently investigate some purely algebraic and combinatorial objects, such as certain classes of symmetric polynomials, Gelfand-Tsetlin patterns, Young tableaux, and sorting networks.

Here and hereNikos Zygouras and I studied some KPZ integrable models on the 2D integer lattice (log-gamma polymer and exponential last passage percolation) with various point-to-(half-)line geometries of directed lattice paths. We found novel connections with special functions from algebraic combinatorics and representation theory, such as orthogonal Whittaker functions and symplectic Schur functions.

In another paper with Nikos Zygouras, we established numerous formulas for last passage percolation models with various symmetries in terms of all the irreducible characters of the classical groups and, more generally, in terms of certain classes of interpolating symmetric polynomials. At the level of KPZ asymptotic analysis, our formulas yielded new routes to universal random matrix limiting distributions and provided a structural explanation of the duality between their Pfaffian and determinant formulations.

In a joint work with Fabio Deelan Cunden, Shane Gibbons, and Dan Romik (see extended abstract and full version), we studied some connections between the oriented swap process, the corner growth model, and the last passage percolation model. We also conjectured a distributional identity between these models that implies an asymptotic result for the absorbing time of the oriented swap process. This conjecture is equivalent to a purely combinatorial identity that involves sorting networks and staircase shape Young tableaux and is connected to the celebrated Edelman-Greene bijection.

In this paper with Neil O’Connell and Nikos Zygouras, we constructed a geometric lifting of the Burge correspondence (a modification of the Robinson-Schensted-Knuth algorithm) as a composition of local birational maps. We applied our construction to a model of two polymer paths of given length constrained to have the same endpoint, known as polymer replica: we found that the distribution of the polymer replica partition function in a log-gamma random environment is a certain Whittaker measure.

Applied cryptography

In 2014 I worked as a research intern at STMicroelectronics, Agrate Brianza (Italy), in the Advanced System Technology security group. I collaborated with Guido Bertoni, Eleonora Cagli, Filippo Melzani, and Ruggero Susella.

I worked on the security of symmetric-key cryptographic algorithms such as AES, with special attention to high order Boolean masking schemes designed to thwart side channel attacks (e.g. Correlation Power Analysis). These 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. My research focused on studying equivalent and easy-to-verify conditions for the vulnerability of leakage combinations, useful to validate the protection order guaranteed by a given cryptographic implementation. I used Matlab and C. I was a referee for ASYACRYPT 2014.

After further elaboration of this research with Filippo Melzani and Vittorio Zaccaria, in 2016 we wrote this article.