Pierre Houdebert
We prove phase transition for the non-symmetric continuum Potts model with background interaction, by generalizing the methods introduced in the symmetric case by Georgii and Häggström. The proof relies on a Fortuin-Kasteleyn representation, percolation and stochastic domination arguments.
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We establish phase transitions for classes of continuum Delaunay multi-type particle systems (continuum Potts models) with infinite range repulsive interaction between particles of different type. In one class of the Delaunay Potts models studied the repulsive interaction is a triangle (multi-body) interaction whereas in the second class the interaction is between pairs (edges) of the Delaunay graph. The result for the edge model is an extension of finite range results in \cite{BBD04} for the Delaunay graph and in \cite{GH96} for continuum Potts models to an infinite range repulsion decaying w…
This is the second of two papers on a continuum version of the Potts model, where particles are points in $\mathbb R^d$, $d\ge 2$, with a spin which may take $S\ge 3$ possible values. Particles with different spins repel each other via a Kac pair potential of range $\ga^{-1}$, $\ga>0$. In this paper we prove phase transition, namely we prove that if the scaling parameter of the Kac potential is suitably small, given any temperature there is a value of the chemical potential such that at the given temperature and chemical potential there exist $S+1$ mutually distinct DLR measures.
In this paper we study a continuum version of the Potts model. Particles are points in R^d, with a spin which may take S possible values, S being at least 3. Particles with different spins repel each other via a Kac pair potential. In mean field, for any inverse temperature there is a value of the chemical potential at which S+1 distinct phases coexist. For each mean field pure phase, we introduce a restricted ensemble which is defined so that the empirical particles densities are close to the mean field values. Then, in the spirit of the Dobrushin Shlosman theory, we get uniqueness and expone…
We provide an explicit formula for the limiting free energy density (log-partition function divided by the number of vertices) for ferromagnetic Potts models on uniformly sparse graph sequences converging locally to the d-regular tree for d even, covering all temperature regimes. This formula coincides with the Bethe free energy functional evaluated at a suitable fixed point of the belief propagation recursion on the d-regular tree, the so-called replica symmetric solution. For uniformly random d-regular graphs we further show that the replica symmetric Bethe formula is an upper bound for the …
The reconstruction problem on the tree has been studied in numerous contexts including statistical physics, information theory and computational biology. However, rigorous reconstruction thresholds have only been established in a small number of models. We prove the first exact reconstruction threshold in a non-binary model establishing the Kesten-Stigum bound for the 3-state Potts model on regular trees of large degree. We further establish that the Kesten-Stigum bound is not tight for the $q$-state Potts model when $q \geq 5$. Moreover, we determine asymptotics for the reconstruction thresho…
We consider the Curie-Weiss Potts model in zero external field under independent symmetric spin-flip dynamics. We investigate dynamical Gibbs-non-Gibbs transitions for a range of initial inverse temperatures beta<3, which covers the phase transition point beta=4 log 2 [8]. We show that finitely many types of trajectories of bad empirical measures appear, depending on the parameter beta, with a possibility of re-entrance into the Gibbsian regime, of which we provide a full description.
We establish phase transitions for continuum Delaunay multi-type particle systems (continuum Potts or Widom-Rowlinson models) with a repulsive interaction between particles of different types. Our interaction potential depends solely on the length of the Delaunay edges. We show that a phase transition occurs for sufficiently large activities and for sufficiently large potential parameter proving an old conjecture of Lebowitz and Lieb extended to the Delaunay structure. Our approach involves a Delaunay random-cluster representation analogous to the Fortuin-Kasteleyn representation of the Potts …
We prove an inequality on decision trees on monotonic measures which generalizes the OSSS inequality on product spaces. As an application, we use this inequality to prove a number of new results on lattice spin models and their random-cluster representations. More precisely, we prove that 1. For the Potts model on transitive graphs, correlations decay exponentially fast for $β<β_c$. 2. For the random-cluster model with cluster weight $q\geq1$ on transitive graphs, correlations decay exponentially fast in the subcritical regime and the cluster-density satisfies the mean-field lower bound in the…
The Quantum Symmetric Simple Exclusion Process in the Continuum and Free Processes
本文提出量子对称简单排斥过程在连续域中的表述，探讨其作为非交换过程的特性，并为宏观波动理论的量子扩展奠定基础。
Diluted mean-field models are spin systems whose geometry of interactions is induced by a sparse random graph or hypergraph. Such models play an eminent role in the statistical mechanics of disordered systems as well as in combinatorics and computer science. In a path-breaking paper based on the non-rigorous `cavity method', physicists predicted not only the existence of a replica symmetry breaking phase transition in such models but also sketched a detailed picture of the evolution of the Gibbs measure within the replica symmetric phase and its impact on important problems in combinatorics, c…
We give a criterion of the form Q(d)c(M)<1 for the non-reconstructability of tree-indexed q-state Markov chains obtained by broadcasting a signal from the root with a given transition matrix M. Here c(M) is an explicit function, which is convex over the set of M's with a given invariant distribution, that is defined in terms of a (q-1)-dimensional variational problem over symmetric entropies. Further Q(d) is the expected number of offspring on the Galton-Watson tree. This result is equivalent to proving the extremality of the free boundary condition-Gibbs measure within the corresponding Gibbs…
We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loop-ensemble CLE$_{κ'}$ for $κ'$ in $(4,8)$ that is drawn on an independent $γ$-LQG surface for $γ^2=16/κ'$. The results are similar in flavor to the ones from our paper dealing with CLE$_κ$ for $κ$ in $(8/3,4)$, where the loops of the CLE are disjoint and simple. In particular, we encode the combined structure of the LQG surface and the CLE$_{κ'}$ in terms of stable growth-fragmentation trees or their variants, which also appear in the asymptotic study of peeling processes on d…
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