On the Description of Macroscopic SystemsGases, liquids and solids are the most familiar examples of large physical systems of particles encountered in nature. In the first part of this introduction, for the sake of simplicity and concreteness, we will mainly consider the system of a gas contained in a vessel.

International audienceWe describe the scaling limit of the nearest neighbour prudent walk on the square lattice, which performs steps uniformly in directions in which it does not see sites already visited. We show that the process eventually settles in one of the quadrants, and derive its scaling limit, which can be expressed in terms of a pair of independent stable subordinators. We also show that the asymptotic speed of the walk is well-defined in the L_1 -norm and equals 3/7. This process possesses unusual properties: it is ballistic but does not have an asymptotic direction, and several natural observables display ageing

At first order phase transition the free energy does not have an analytic continuation in the thermodynamical variable, which is conjugate to an order parameter for the transition. This result is proved at low temperature for lattice models with finite range interaction and two periodic ground-states, under the only condition that they verify Peierls condition.

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In this note we consider long range $q$-states Potts models on $\mathbf{Z}^d$, $d\geq 2$. For various families of non-summable ferromagnetic pair potentials $\phi(x)\geq 0$, we show that there exists, for all inverse temperature $\beta>0$, an integer $N$ such that the truncated model, in which all interactions between spins at distance larger than $N$ are suppressed, has at least $q$ distinct infinite-volume Gibbs states. This holds, in particular, for all potentials whose asymptotic behaviour is of the type $\phi(x)\sim \|x\|^{-\alpha}$, $0\leq\alpha\leq d$. These results are obtained using simple percolation arguments.Comment: 18 pages, 4 figure