Research

• An algorithm for geo-distributed and redundant storage in Garage, with Alex Auvolat, preprint
  This paper presents an optimal algorithm to compute the assignment of data to storage nodes in the Garage geo-distributed storage system. We discuss the complexity of the different steps of the algorithm and metrics that can be displayed to the user.
• Random geometry and free energy of critical planar lattice models, PhD thesis
  In addition to results on rotational invariance of FK percolation and delocalization of 6V height function, we expose in a bibliographical section the rapidity line formalism of isoradial graphs and the q-deformed Knizhnik–Zamolodchikov equations.
• Delocalization of the height function of the six-vertex model, with Hugo Duminil-Copin, Alex Karrila, Ioan Manolescu, preprint
  We show that the height function of the six-vertex model, in the parameter range $a=b=1$ and $c\ge 1$, is delocalized with logarithmic variance when $c\le 2$. This complements the earlier proven localization for $c>2$. Our proof relies on Russo--Seymour--Welsh type arguments, and on the local behaviour of the free energy of the cylindrical six-vertex model, as a function of the unbalance between the number of up and down arrows.
• Rotational invariance in critical planar lattice models, with Hugo Duminil-Copin, Karol Kajetan Kozlowski, Dmitry Krachun, Ioan Manolescu, preprint
  In this paper, we prove that the large scale properties of a number of two-dimensional lattice models are rotationally invariant. More precisely, we prove that the random-cluster model on the square lattice with cluster-weight $1\le q\le 4$ exhibits rotational invariance at large scales. This covers the case of Bernoulli percolation on the square lattice as an important example. We deduce from this result that the correlations of the Potts models with $q\in\{2,3,4\}$ colors and of the six-vertex height function with $\Delta\in[-1,-1/2]$ are rotationally invariant at large scales.
• Replica Bounds by Combinatorial Interpolation for Diluted Spin Systems, with Marc Lelarge, J Stat Phys 173, 917–940 (2018)
  In two papers Franz, Leone and Toninelli proved bounds for the free energy of diluted random constraints satisfaction problems, for a Poisson degree distribution and a general distribution. Panchenko and Talagrand simplified the proof and generalized the result of for the Poisson case. We provide a new proof for the general degree distribution case and as a corollary, we obtain new bounds for the size of the largest independent set (also known as hard core model) in a large random regular graph. Our proof uses a combinatorial interpolation based on biased random walks and allows to bypass the arguments in based on the study of the Sherrington-Kirkpatrick (SK) model.
• Abstract Interpretation with Higher-Dimensional Ellipsoids and Conic Extrapolation, with Arnaud Venet, CAV 2015: 415-430
  The inference and the verification of numerical relationships among variables of a program is one of the main goals of static analysis. In this paper, we propose an Abstract Interpretation framework based on higher-dimensional ellipsoids to automatically discover symbolic quadratic invariants within loops, using loop counters as implicit parameters. In order to obtain non-trivial invariants, the diameter of the set of values taken by the numerical variables of the program has to evolve (sub-)linearly during loop iterations. These invariants are called ellipsoidal cones and can be seen as an extension of constructs used in the static analysis of digital filters. Semidefinite programming is used to both compute the numerical results of the domain operations and provide proofs (witnesses) of their correctness.