Click here for the French version.



Hello! I am Hugo Vanneuville, I am a researcher in math. Since January 2020, I am a CNRS researcher at Institut Fourier (Université Grenoble-Alpes). Before this, I have completed a PhD at Institut Camille Jordan (Université Lyon 1) under the direction of Christophe Garban and I have been a post-doctoral resarcher at the math department of ETH Zürich under the direction of Vincent Tassion.


Research interests:

I am interested in phase transition and noise sensitivity phenomenons in percolation models and in the universality properties of these models (in 2D in most of my works). In percolation theory, one studies the connectivity properties of random sets (such as a random subset of edges of a given lattice, or (sub)level sets of a random function). When the density of the random set reaches some critical value, fractal structures appear; they are the sign of the emergence of interactions at all scales.

More specifically, the following three questions are guiding my research (among others):
  • Can we find new ways of formalising the fact that a threshold effect appears if the event that we study "depends little on every given coordinate"?
  • How can we show noise sensitivity properties without using any spectral tool?
  • Can we prove universality properties by using noise sensitivity?

    • I study or use the following mathematical objects: boundaries of events, their spectrum, random interfaces, scaling relations, concentration of measure inequalities, inequalities about the influences...

    (See also here for a description of my research interests and other things - in French.)

    Photo
    Figures: Percolation on a random tiling: each face is colored black with probability p=0.4, 0.5 and 0.6 respectively. In red: the points connected to the left side by a black path. There is a transition at p=0.5.


    Research papers:

    I have written papers with Hugo Duminil-Copin, Christophe Garban, Laurin Köhler-Schindler, Stephen Muirhead, Alejandro Rivera, Pierre-François Rodriguez and Vincent Tassion.

    Noise sensitivity and sharp thresholds for Bernoulli percolation:

  • [iii] Sharpness of Bernoulli percolation via couplings     Arxiv   (preprint)
  • [ii] Noise sensitivity of percolation via differential inequalities,     with Vincent Tassion   Arxiv   (preprint)
  • [i] Exceptional times for percolation under exclusion dynamics,     with Christophe Garban   Arxiv   (in Annales scientifiques de l'École normale supérieure, 2019)

    • Random nodal lines:

  • [vi] Existence of an unbounded nodal hypersurface for smooth Gaussian fields in dimension $d \ge 3$,     with Hugo Duminil-Copin, Alejandro Rivera and Pierre-François Rodriguez   Arxiv   (preprint)
  • [v] The phase transition for planar Gaussian percolation models without FKG,     with Stephen Muirhead and Alejandro Rivera, and an appendix by Laurin Köhler-Schindler   Arxiv   (preprint)
  • [iv] Bargmann-Fock percolation is noise sensitive,     with Christophe Garban   Arxiv   (in Electronic Journal of Probability, 2020)
  • [iii] The sharp phase transition for level set percolation of smooth planar Gaussian fields,     with Stephen Muirhead   Arxiv   (in Annales de l'Institut Henri Poincaré, prob. et stat., 2020)
  • [ii] The critical threshold for Bargmann-Fock percolation,     with Alejandro Rivera   Arxiv   (in Annales Henri Lebesgue, 2020)
  • [i] Quasi-independence for nodal lines,     with Alejandro Rivera   Arxiv   (in Annales de l'Institut Henri Poincaré, prob. et stat., 2019)

    • Voronoi percolation:

  • [iii] The annealed spectral sample of Voronoi percolation     Arxiv   (in Annals of Probability, 2021)
  • [ii] Quantitative quenched Voronoi percolation and applications     Arxiv   (preprint)
  • [i] Annealed scaling relations for Voronoi percolation     Arxiv   (in Electronic Journal of Probability, 2019)


    • Expository article:

    I am writing an expository article about phase transition and noise sensitivity for Boolean functions, but it takes more time than I expected! Meanwhile, you can find here some material that might be of some interest.

  • Noise sensitivity without spectrum on a simple example:     Pdf


    • Probability seminar at Institut Fourier:

    I'm organizing the probability seminar at Institut Fourier. More information here.


    Teaching: (in French) Enseignement en M1 Maths Générales à l'Université Grenoble-Alpes:

    Pendant le second semestre de l'année 2021-2022, j'enseigne avec Agnès Coquio le cours "Processus de Markov" en M1MG à l'Université de Grenoble. Le cours est divisé en quatre parties. Cette année, je m'occupe des Parties II et IV.

  • Partie I : Rappels et compléments sur les chaînes de Markov
  • Partie II : Temps de mélange pour les chaînes de Markov
  • Partie III : Chaînes de Markov à temps continu
  • Partie IV : Introduction au mouvement brownien


    • A counterexample?

    Consider the hypercube {0,1}^n equipped with the product probability measures of parameter p, that we denote by P_p. Let A be an increasing subset of {0,1}^n. By the works of Russo, BKKKL and Talagrand, if max_{i,p} P_p [ changing the i^th coordinate changes 1_A ] is small, then A satisfies a sharp threshold in the sense that P_p[A] is either close to 0 or close to 1 except when p belongs to some small interval. Now, let q such that P_q[A]=1/2. Is it still true if we only assume that max_i P_q [ changing the i^th coordinate changes 1_A ] is small?

    Another question: once again, let us fix some parameter q such that max_i P_q [ changing the i^th coordinate changes 1_A ] is small. Is P_q [ the number of pivotal points lies between 1 and 10 ] necessarily small?


    Slides and videos:

    (Link to the webpage of Percolation Today.)

    Videos of talks at Percolation Today:

  • You can find here a talk about the geometry of random nodal surfaces in dimension 3.
  • You can find here a talk with Vincent Tassion about noise sensitivity via differential inequalities.
  • You can find here a talk with Stephen Muirhead about sharp threshold phenomenons for planar Gaussian percolation without FKG.

    • Slides: (some are in French)

  • Noise sensitivity without spectrum and covariance formulas: Oberwolfach online, Perco today
  • Sharp threshold phenomenons and percolation without FKG: Luxembourg online, Perco today, Zürich online
  • Percolation and random nodal lines: Orsay 2, Buda, Genève, Orsay 1, Saint-Flour, Oxford
  • Dynamical percolation, Fourier spectrum and percolation in random environment: Pest, Göteborg, Aussois, Zürich, Saint-Flour, Marseille


    • PhD thesis: (the introduction is written in French and the chapters are written in English)

    I have defended my PhD thesis on November 28th, 2018.     Pdf     Slides (in French)     Small erratum


    Reading group in Zürich:

    You can find here a link to the webpage of the reading group on nodal lines that took place in autumn 2019 at ETHZ and Zürich University. I have written an introductory text for this reading group (with a focus on concentration phenomenons). You can find this text below:

    Overview and concentration results for nodal lines :     Pdf


    Diffusion: (in French)

    Exposé lors du "Barcamp" organisé par les bibliothécaires de Lyon 1 en mai 2018. Vous pouvez trouver la vidéo ici.