57 research outputs found
A universality property for last-passage percolation paths close to the axis
We consider a last-passage directed percolation model in , with i.i.d.
weights whose common distribution has a finite th moment. We study the
fluctuations of the passage time from the origin to the point
. We show that, for suitable (depending
on ), this quantity, appropriately scaled, converges in distribution as
to the Tracy-Widom distribution, irrespective of the underlying
weight distribution. The argument uses a coupling to a Brownian directed
percolation problem and the strong approximation of Koml\'os, Major and
Tusn\'ady.Comment: 8 page
Universal current fluctuations in the symmetric exclusion process and other diffusive systems
We show, using the macroscopic fluctuation theory of Bertini, De Sole,
Gabrielli, Jona-Lasinio, and Landim, that the statistics of the current of the
symmetric simple exclusion process (SSEP) connected to two reservoirs are the
same on an arbitrary large finite domain in dimension as in the one
dimensional case. Numerical results on squares support this claim while results
on cubes exhibit some discrepancy. We argue that the results of the macroscopic
fluctuation theory should be recovered by increasing the size of the contacts.
The generalization to other diffusive systems is straightforward.Comment: 6 pages, 4 figure
The Brownian motion as the limit of a deterministic system of hard-spheres
We provide a rigorous derivation of the brownian motion as the limit of a
deterministic system of hard-spheres as the number of particles goes to
infinity and their diameter simultaneously goes to , in the
fast relaxation limit (with a suitable
diffusive scaling of the observation time). As suggested by Hilbert in his
sixth problem, we rely on a kinetic formulation as an intermediate level of
description between the microscopic and the fluid descriptions: we use indeed
the linear Boltzmann equation to describe one tagged particle in a gas close to
global equilibrium. Our proof is based on the fundamental ideas of Lanford. The
main novelty here is the detailed study of the branching process, leading to
explicit estimates on pathological collision trees
Spectral Gap Critical Exponent for Glauber Dynamics of Hierarchical Spin Models
Abstract: We develop a renormalisation group approach to deriving the asymptotics of the spectral gap of the generator of Glauber type dynamics of spin systems with strong correlations (at and near a critical point). In our approach, we derive a spectral gap inequality for the measure recursively in terms of spectral gap inequalities for a sequence of renormalised measures. We apply our method to hierarchical versions of the 4-dimensional n-component |Ï|4 model at the critical point and its approach from the high temperature side, and of the 2-dimensional Sine-Gordon and the Discrete Gaussian models in the rough phase (KosterlitzâThouless phase). For these models, we show that the spectral gap decays polynomially like the spectral gap of the dynamics of a free field (with a logarithmic correction for the |Ï|4 model), the scaling limit of these models in equilibrium
From Hard Sphere Dynamics to the StokesâFourier Equations: An Analysis of the BoltzmannâGrad Limit
to appear, Annals of PDEsInternational audienceWe derive the linear acoustic and Stokes-Fourier equations as the limiting dynamics of a system of N hard spheres of diameter in two space dimensions, when N , 0, N = , using the linearized Boltzmann equation as an intermediate step. Our proof is based on Lanford's strategy [18], and on the pruning procedure developed in [5] to improve the convergence time to all kinetic times with a quantitative control which allows us to reach also hydrodynamic time scales. The main novelty here is that uniform L 2 a pri-ori estimates combined with a subtle symmetry argument provide a weak version of chaos, in the form of a cumulant expansion describing the asymptotic decorrelation between the particles. A refined geometric analysis of recollisions is also required in order to discard the possibility of multiple recollisions
LogâSobolev Inequality for the Continuum SineâGordon Model
We derive a multiscale generalisation of the BakryâĂmery criterion for a measure to satisfy a logâSobolev inequality. Our criterion relies on the control of an associated PDE wellâknown in renormalisation theory: the Polchinski equation. It implies the usual BakryâĂmery criterion, but we show that it remains effective for measures that are far from logâconcave. Indeed, using our criterion, we prove that the massive continuum sineâGordon model with ÎČ < 6Ï satisfies asymptotically optimal logâSobolev inequalities for Glauber and Kawasaki dynamics. These dynamics can be seen as singular SPDEs recently constructed via regularity structures, but our results are independent of this theory. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC
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