arXiv subjects

Quentin Moulard

Publications and source records attributed to Quentin Moulard.

3 recordsLinked to original sources

arXiv2026

Superdiffusive central limit theorem for a class of driven diffusive systems at the critical dimension

We study the large-scale behaviour of a class of driven diffusive systems modelled by a Stochastic Partial Differential Equation, the Stochastic Burgers Equation (SBE) with general nonlinearity, at the critical dimension and in infinite volume. Our main result shows that, under a logarithmically superdiffusive space-time scaling, it is given by the same explicit Gaussian Fixed point obtained in [G. Cannizzaro, Q. Moulard, & F. Toninelli, arxiv.org/abs/2501.00344, 2025] for the quadratic SBE, but with suitably renormalised coefficients, thereby rigorously justifying and partly correcting the classical Physics derivation of the SBE in [H. van Beijeren, R. Kutner, & H. Spohn, Phys. Rev. Lett., 1986] based on Spohn's theory of nonlinear fluctuating hydrodynamics. Besides, ours is the first universality-type result for out-of-equilibrium systems and the first extension of [M. Hairer, J. Quastel, Forum of Mathematics, Pi, Vol. 6, 2018, e3], to the critical dimension and beyond weak coupling. The major challenge in our work is the mild growth condition on the nonlinearity which renders even the well-posedness of the microscopic equation non-trivial. Additional key novelties include the derivation of fine estimates on the non-quadratic part of the generator as well as a new approximation for the resolvent associated to the solution of the quadratic SBE.

math.PR↗

arXiv2025

Dimers with layered disorder

We study the dimer model on the square grid, with quenched random edge weights. Randomness is chosen to have a layered structure, similar to that of the celebrated McCoy-Wu disordered Ising model. Disorder has a highly non-trivial effect and it produces an essential singularity of the free energy, with $e^{-\sqrt{{\rm distance}}}$ decay of dimer-dimer correlations, at a point of the ``liquid'' (or ``massless'') phase where the homogeneous dimer model has instead a real analytic free energy and correlations decaying like $1/({\rm distance})^2$. Moreover, at a point where the homogeneous model has a transition between a massive (gaseous) and massless (liquid) phase, the critical exponent 3/2 (Pokrovsky-Talapov law), characteristic of the transition between the two regimes, is modified by disorder into an exponent that ranges continuously between 3/2 and infinity.

math.PR↗

arXiv2024

Superdiffusive Central Limit Theorem for the Stochastic Burgers Equation at the critical dimension

The Stochastic Burgers Equation (SBE) is a singular, non-linear Stochastic Partial Differential Equation (SPDE) that describes, on mesoscopic scales, the fluctuations of stochastic driven diffusive systems with a conserved scalar quantity. In space dimension d = 2, the SBE is critical, being formally scale invariant under diffusive scaling. As such, it falls outside of the domain of applicability of the theories of Regularity Structures and paracontrolled calculus. In apparent contrast with the formal scale invariance, we fully prove the conjecture first appeared in [H. van Beijeren, R. Kutner, & H. Spohn, Phys. Rev. Lett., 1986] according to which the 2d-SBE is logarithmically superdiffusive, i.e. its diffusion coefficient diverges like $(\log t)^{2/3}$ as $t\to\infty$, thus removing subleading diverging multiplicative corrections in [D. De Gaspari & L. Haunschmid-Sibitz, Electron. J. Probab., 2024] and in [H.-T. Yau, Ann. of Math., 2004] for 2d-ASEP. We precisely identify the constant prefactor of the logarithm and show it is proportional to $\lambda^{4/3}$, for $\lambda>0$ the coupling constant, which, intriguingly, turns out to be exactly the same as for the one-dimensional Stochastic Burgers/KPZ equation. More importantly, we prove that, under super-diffusive space-time rescaling, the SBE has an explicit Gaussian fixed point in the Renormalization Group sense, by deriving a superdiffusive central limit-type theorem for its solution. This is the first scaling limit result for a critical singular SPDE, beyond the weak coupling regime, and is obtained via a refined control, on all length-scales, of the resolvent of the generator of the SBE. We believe our methods are well-suited to study other out-of-equilibrium driven diffusive systems at the critical dimension, such as 2d-ASEP, which, we conjecture, have the same large-scale Fixed Point as SBE.

math.PR↗