Searcharxiv⌕ Search

arXiv subjects

Jean Ruiz

Publications and source records attributed to Jean Ruiz.

18 recordsLinked to original sources
arXiv2017

Glassy states: the free Ising model on a tree

We consider the ferromagnetic Ising model on the Cayley tree and we investigate the decomposition of the free state into extremal states below the spin glass temperature. We show that this decomposition has uncountably many components. The tail observable showing that the free state is not extremal is related to the Edwards-Anderson parameter, measuring the variance of the (random) magnetization obtained from drawing boundary conditions from the free state.

cond-mat.stat-mech↗
arXiv2015

Geometric expansion of the log-partition function of the anisotropic Heisenberg model

We study the asymptotic expansion of the log-partition function of the anisotropic Heisenberg model in a bounded domain as this domain is dilated to infinity. Using the Ginibre's representation of the anisotropic Heisenberg model as a gas of interacting trajectories of a compound Poisson process we find all the non-decreasing terms of this expansion. They are given explicitly in terms of functional integrals. As the main technical tool we use the cluster expansion method.

math-ph↗
arXiv2012

On the Wulff construction as a problem of equivalence of statistical ensembles

The statistical mechanics of SOS (solid-on-solid) 1-dimensional models under the global constraint of having a specified area between the interface and the horizontal axis, is studied. We prove the existence of the thermodynamic limits and the equivalence of the corresponding statistical mechanics. This gives a simple alternative microscopic proof of the validity of the Wulff construction for such models.

cond-mat.stat-mech↗
arXiv2008

Limit Theorems and Coexistence Probabilities for the Curie-Weiss Potts Model with an external field

The Curie-Weiss Potts model is a mean field version of the well-known Potts model. In this model, the critical line $β= β_c (h)$ is explicitly known and corresponds to a first order transition when $q > 2$. In the present paper we describe the fluctuations of the density vector in the whole domain $β\geqslant 0$ and $h \geqslant 0$, including the conditional fluctuations on the critical line and the non-Gaussian fluctuations at the extremity of the critical line. The probabilities of each of the two thermodynamically stable states on the critical line are also computed. Similar results are inferred for the Random-Cluster model on the complete graph.

math.PR↗
arXiv2008

Lower Spectral Branches of a Spin-Boson Model

We study the structure of the spectrum of a two-level quantum system weakly coupled to a boson field (spin-boson model). Our analysis allows to avoid the cutoff in the number of bosons, if their spectrum is bounded below by a positive constant. We show that, for small coupling constant, the lower part of the spectrum of the spin-boson Hamiltonian contains (one or two) isolated eigenvalues and (respectively, one or two) manifolds of atom $+ 1$-boson states indexed by the boson momentum $q$. The dispersion laws and generalized eigenfunctions of the latter are calculated.

cond-mat.stat-mech↗
arXiv2008

On the Kertész line: Thermodynamic versus Geometric Criticality

The critical behaviour of the Ising model in the absence of an external magnetic field can be specified either through spontaneous symmetry breaking (thermal criticality) or through cluster percolation (geometric criticality). We extend this to finite external fields for the case of the Potts' model, showing that a geometric analysis leads to the same first order/second order structure as found in thermodynamic studies. We calculate the Kertész line, separating percolating and non-percolating regimes, both analytically and numerically for the Potts model in presence of an external magnetic field.

cond-mat.stat-mech↗
arXiv2008

Thermodynamic versus Topological Phase Transitions: Cusp in the Kertész Line

We present a study of phase transitions of the Curie--Weiss Potts model at (inverse) temperature $β$, in presence of an external field $h$. Both thermodynamic and topological aspects of these transitions are considered. For the first aspect we complement previous results and give an explicit equation of the thermodynamic transition line in the $β$--$h$ plane as well as the magnitude of the jump of the magnetization (for $q \geqslant 3)$. The signature of the latter aspect is characterized here by the presence or not of a giant component in the clusters of a Fortuin--Kasteleyn type representation of the model. We give the equation of the Kertész line separating (in the $β$--$h$ plane) the two behaviours. As a result, we get that this line exhibits, as soon as $q \geqslant 3$, a very interesting cusp where it separates from the thermodynamic transition line.

cond-mat.stat-mech↗
arXiv2008

On the Kertész line: Some rigorous bounds

We study the Kertész line of the $q$--state Potts model at (inverse) temperature $β$, in presence of an external magnetic field $h$. This line separates two regions of the phase diagram according to the existence or not of an infinite cluster in the Fortuin-Kasteleyn representation of the model. It is known that the Kertész line $h_K (β)$ coincides with the line of first order phase transition for small fields when $q$ is large enough. Here we prove that the first order phase transition implies a jump in the density of the infinite cluster, hence the Kertész line remains below the line of first order phase transition. We also analyze the region of large fields and prove, using techniques of stochastic comparisons, that $h_K (β)$ equals $\log (q - 1) - \log (β- β_p)$ to the leading order, as $β$ goes to $β_p = - \log (1 - p_c)$ where $p_c$ is the threshold for bond percolation.

cond-mat.stat-mech↗
arXiv2007

On a model of random cycles

We consider a model of random permutations of the sites of the cubic lattice. Permutations are weighted so that sites are preferably sent onto neighbors. We present numerical evidence for the occurrence of a transition to a phase with infinite, macroscopic cycles.

cond-mat.stat-mech↗
arXiv2006

On the mean Euler characteristic and mean Betti numbers of the Ising model with arbitrary spin

The behaviour of the mean Euler-Poincaré characteristic and mean Betti's numbers in the Ising model with arbitrary spin on $\mathbbm{Z}^2$ as functions of the temperature is investigated through intensive Monte Carlo simulations. We also consider these quantities for each color $a$ in the state space $S\_Q = \{- Q, - Q + 2, ..., Q \}$ of the model. We find that these topological invariants show a sharp transition at the critical point.

cond-mat.stat-mech↗
arXiv2006

A lattice model for the line tension of a sessile drop

Within a semi--infinite thre--dimensional lattice gas model describing the coexistence of two phases on a substrate, we study, by cluster expansion techniques, the free energy (line tension) associated with the contact line between the two phases and the substrate. We show that this line tension, is given at low temperature by a convergent series whose leading term is negative, and equals 0 at zero temperature.

cond-mat.stat-mech↗
arXiv2005

On the Surface Tensions of Binary Mixtures

For binary mixtures with fixed concentrations of the species, various relationships between the surface tensions and the concentrations are briefly reviewed.

cond-mat.stat-mech↗
arXiv1999

Is there an Optimal Substrate Geometry for Wetting ?

We consider the problem of the Winterbottom's construction and Young's equation in the presence of a rough substate and establish their microscopic validity within a 1+1-dimensional SOS type model. We then present the low temperature expansion of the wall tension leading to the Wenzel's law for the wall tension and its corrections. Finally, for a fix roughness, we compare the influence of different geometries of the substrate on wetting properties. We show that there is an optimal geometry with a given roughness for a certain class of simple substrates. Our results are in agreement and explain recent numerical simulations.

cond-mat↗