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Register a new userLower bound on the mean square displacement of particles in the hard disk model
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Thomas Richthammer
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The hard disk model is a 2D Gibbsian process of particles interacting via pure hard core repulsion. At high particle density the model is believed to show orientational order, however, it is known not to exhibit positional order. Here we investigate to what extent particle positions may fluctuate. We consider a finite volume version of the model in a box of dimensions $2n times 2n$ with arbitrary boundary configuration,and we show that the mean square displacement of particles near the center of the box is bounded from below by $c log n$. The result generalizes to a large class of models with fairly arbitrary interaction.
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While 2D Gibbsian particle systems might exhibit orientational order resulting in a lattice-like structure, these particle systems do not exhibit positional order if the interaction between particles satisfies some weak assumptions. Here we investigate to which extent particles within a box of size $2n times 2n$ may fluctuate from their ideal lattice position. We show that particles near the center of the box typically show a displacement at least of order $sqrt{log n}$. Thus we extend recent results on the hard disk model to particle systems with fairly arbitrary particle spins and interaction. Our result applies to models such as rather general continuum Potts type models, e.g. with Widom-Rowlinson or Lenard-Jones-type interaction.
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We consider the mean-field classical Heisenberg model and obtain detailed information about the total spin of the system by studying the model on a complete graph and sending the number of vertices to infinity. In particular, we obtain Cramer- and Sanov-type large deviations principles for the total spin and the empirical spin distribution and demonstrate a second-order phase transition in the Gibbs measures. We also study the asymptotics of the total spin throughout the phase transition using Steins method, proving central limit theorems in the sub- and supercritical phases and a nonnormal limit theorem at the critical temperature.
We derive mean-field equations for a general class of ferromagnetic spin systems with an explicit error bound in finite volumes. The proof is based on a link between the mean-field equation and the free convolution formalism of random matrix theory, which we exploit in terms of a dynamical method. We present three sample applications of our results to Ka{c} interactions, randomly diluted models, and models with an asymptotically vanishing external field.
We consider the large-time behavior of the solution $ucolon [0,infty)timesZto[0,infty)$ to the parabolic Anderson problem $partial_t u=kappaDelta u+xi u$ with initial data $u(0,cdot)=1$ and non-positive finite i.i.d. potentials $(xi(z))_{zinZ}$. Unlike in dimensions $dge2$, the almost-sure decay rate of $u(t,0)$ as $ttoinfty$ is not determined solely by the upper tails of $xi(0)$; too heavy lower tails of $xi(0)$ accelerate the decay. The interpretation is that sites $x$ with large negative $xi(x)$ hamper the mass flow and hence screen off the influence of more favorable regions of the potential. The phenomenon is unique to $d=1$. The result answers an open question from our previous study cite{BK00} of this model in general dimension.
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