Results for "Hugo Duminil-Copin"

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Upper bounds on the percolation correlation lengthFeb 08 2019We study the size of the near-critical window for Bernoulli percolation on $\mathbb Z^d$. More precisely, we use a quantitative Grimmett-Marstrand theorem to prove that the correlation length, both below and above criticality, is bounded from above by ... More
The box-crossing property for critical two-dimensional oriented percolationOct 31 2016We consider critical oriented Bernoulli percolation on the square lattice $\mathbb{Z}^2$. We prove a Russo-Seymour-Welsh type result which allows us to derive several new results concerning the critical behavior: - We establish that the probability that ... More
Exponential decay of loop lengths in the loop $O(n)$ model with large $n$Dec 29 2014Nov 01 2015The loop $O(n)$ model is a model for a random collection of non-intersecting loops on the hexagonal lattice, which is believed to be in the same universality class as the spin $O(n)$ model. It has been conjectured that both the spin and the loop $O(n)$ ... More
Macroscopic loops in the loop $O(n)$ model at Nienhuis' critical pointJul 28 2017Nov 16 2017The loop $O(n)$ model is a model for a random collection of non-intersecting loops on the hexagonal lattice, which is believed to be in the same universality class as the spin $O(n)$ model. It has been predicted by Nienhuis that for $0\le n\le 2$ the ... More
Exponential decay of loop lengths in the loop $O(n)$ model with large $n$Dec 29 2014Oct 27 2016The loop $O(n)$ model is a model for a random collection of non-intersecting loops on the hexagonal lattice, which is believed to be in the same universality class as the spin $O(n)$ model. It has been conjectured that both the spin and the loop $O(n)$ ... More
The Bethe ansatz for the six-vertex and XXZ models: an expositionNov 29 2016In this paper, we review a few known facts on the coordinate Bethe ansatz. We present a detailed construction of the Bethe ansatz vector $\psi$ and energy $\Lambda$, which satisfy $V \psi = \Lambda \psi$, where $V$ is the the transfer matrix of the six-vertex ... More
Existence of phase transition for percolation using the Gaussian Free FieldJun 20 2018In this paper, we prove that Bernoulli percolation on bounded degree graphs with isoperimetric dimension $d>4$ undergoes a non-trivial phase transition (in the sense that $p_c<1$). As a corollary, we obtain that the critical point of Bernoulli percolation ... More
Sharp threshold phenomena in statistical physicsOct 08 2018This text describes the content of the Takagi lectures given by the author in Kyoto in 2017. The lectures present some aspects of the theory of sharp thresholds for boolean functions and its application to the study of phase transitions in statistical ... More
Lectures on the Ising and Potts models on the hypercubic latticeJul 03 2017Phase transitions are a central theme of statistical mechanics, and of probability more generally. Lattice spin models represent a general paradigm for phase transitions in finite dimensions, describing ferromagnets and even some fluids (lattice gases). ... More
Random currents expansion of the Ising modelJul 23 2016Jul 14 2017Critical behavior at an order/disorder phase transition has been a central object of interest in statistical physics. In the past century, techniques borrowed from many different fields of mathematics (Algebra, Combinatorics, Probability, Complex Analysis, ... More
Divergence of the correlation length for critical planar FK percolation with $1\le q\le4$ via parafermionic observablesAug 18 2012Sep 23 2012Parafermionic observables were introduced by Smirnov for planar FK percolation in order to study the critical phase $(p,q)=(p_c(q),q)$. This article gathers several known properties of these observables. Some of these properties are used to prove the ... More
Sixty years of percolationDec 13 2017Percolation models describe the inside of a porous material. The theory emerged timidly in the middle of the twentieth century before becoming one of the major objects of interest in probability and mathematical physics. The golden age of percolation ... More
Random currents expansion of the Ising modelJul 23 2016Critical behavior at an order/disorder phase transition has been a central object of interest in statistical physics. In the past century, techniques borrowed from many different fields of mathematics (Algebra, Combinatorics, Probability, Complex Analysis, ... More
Limit of the Wulff Crystal when approaching criticality for site percolation on the triangular latticeMay 26 2013The understanding of site percolation on the triangular lattice progressed greatly in the last decade. Smirnov proved conformal invariance of critical percolation, thus paving the way for the construction of its scaling limit. Recently, the scaling limit ... More
Sharp metastability threshold for an anisotropic bootstrap percolation modelOct 22 2010Mar 30 2016Bootstrap percolation models have been extensively studied during the two past decades. In this article, we study the following "anisotropic" bootstrap percolation model: the neighborhood of a point (m,n) is the set \[\{(m+2,n),(m+1,n),(m,n+1),(m-1,n),(m-2,n),(m,n-1)\}.\] ... More
The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt2}$Jul 04 2010Jun 27 2011We provide the first mathematical proof that the connective constant of the hexagonal lattice is equal to $\sqrt{2+\sqrt 2}$. This value has been derived non rigorously by B. Nienhuis in 1982, using Coulomb gas approach from theoretical physics. Our proof ... More
Planar percolation with a glimpse of Schramm-Loewner EvolutionJul 01 2011Jun 07 2013In recent years, important progress has been made in the field of two-dimensional statistical physics. One of the most striking achievements is the proof of the Cardy-Smirnov formula. This theorem, together with the introduction of Schramm-Loewner Evolution ... More
On the double random current nesting fieldDec 06 2017Mar 20 2019We relate the planar random current representation introduced by Griffiths, Hurst and Sherman to the dimer model. More precisely, we provide a measure-preserving map between double random currents (obtained as the sum of two independent random currents) ... More
The truncated correlations of the Ising model in any dimension decay exponentially fast at all but the critical temperatureJun 01 2015Sep 22 2015The truncated two-point function of the nearest-neighbor ferromagnetic Ising model on $\mathbb Z^d$ ($d\ge3$) in its pure phases is proven to decays exponentially fast throughout the ordered regime ($T<T_c$). Together with known results, this implies ... More
The self-dual point of the two-dimensional random-cluster model is critical for $q\geq 1$Jun 25 2010Nov 27 2013We prove a long-standing conjecture on random-cluster models, namely that the critical point for such models with parameter $q\geq1$ on the square lattice is equal to the self-dual point $p_{sd}(q) = \sqrt q /(1+\sqrt q)$. This gives a proof that the ... More
Renormalization of crossing probabilities in the planar random-cluster modelJan 24 2019The study of crossing probabilities - i.e. probabilities of existence of paths crossing rectangles - has been at the heart of the theory of two-dimensional percolation since its beginning. They may be used to prove a number of results on the model, including ... More
Bridge Decomposition of Restriction MeasuresSep 01 2009Jul 03 2010Motivated by Kesten's bridge decomposition for two-dimensional self-avoiding walks in the upper half plane, we show that the conjectured scaling limit of the half-plane SAW, the SLE(8/3) process, also has an appropriately defined bridge decomposition. ... More
A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising modelFeb 10 2015Jan 21 2018We provide a new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. The proof applies to infinite range models on arbitrary locally finite transitive infinite graphs. For Bernoulli percolation, we prove finiteness ... More
The phase transitions of the planar random-cluster and Potts models with q larger than 1 are sharpSep 12 2014We prove that random-cluster models with q larger than 1 on a variety of planar lattices have a sharp phase transition, that is that there exists some parameter p_c below which the model exhibits exponential decay and above which there exists a.s. an ... More
A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising modelFeb 10 2015Mar 21 2016We provide a new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. The proof applies to infinite range models on arbitrary locally finite transitive infinite graphs. For Bernoulli percolation, we prove finiteness ... More
Conformal invariance of lattice modelsSep 07 2011Jun 21 2012These lecture notes provide a (almost) self-contained account on conformal invariance of the planar critical Ising and FK-Ising models. They present the theory of discrete holomorphic functions and its applications to planar statistical physics (more ... More
On the double random current nesting fieldDec 06 2017We relate the planar random current representation introduced by Griffiths, Hurst and Sherman to the dimer model. More precisely, we provide a measure-preserving map between double random currents (obtained as the sum of two independent random currents) ... More
The critical temperature for the Ising model on planar doubly periodic graphsSep 05 2012We provide a simple characterization of the critical temperature for the Ising model on an arbitrary planar doubly periodic weighted graph. More precisely, the critical inverse temperature \beta for a graph G with coupling constants (J_e)_{e\in E(G)} ... More
Self-avoiding walk is sub-ballisticMay 02 2012We prove that self-avoiding walk on Z^d is sub-ballistic in any dimension d at least two. That is, writing ||u|| for the Euclidean norm of u \in Z^d, and SAW_n for the uniform measure on self-avoiding walks gamma:{0,...,n} \to Z^d for which gamma_0 = ... More
A new proof of the sharpness of the phase transition for Bernoulli percolation on $\mathbb Z^d$Feb 10 2015We provide a new proof of the sharpness of the phase transition for nearest-neighbour Bernoulli percolation. More precisely, we show that - for $p<p_c$, the probability that the origin is connected by an open path to distance $n$ decays exponentially ... More
A note on Schramm's locality conjecture for random-cluster modelsJul 24 2017Aug 30 2017In this note, we discuss a generalization of Schramm's locality conjecture to the case of random-cluster models. We give some partial (modest) answers, and present several related open questions. Our main result is to show that the critical inverse temperature ... More
The near-critical planar FK-Ising modelNov 01 2011Feb 13 2014We study the near-critical FK-Ising model. First, a determination of the correlation length defined via crossing probabilities is provided. Second, a phenomenon about the near-critical behavior of FK-Ising is highlighted, which is completely missing from ... More
Random Currents and Continuity of Ising Model's Spontaneous MagnetizationNov 08 2013Jun 29 2014The spontaneous magnetization is proved to vanish continuously at the critical temperature for a class of ferromagnetic Ising spin systems which includes the nearest neighbor ferromagnetic Ising spin model on $\mathbb Z^d$ in $d=3$ dimensions. The analysis ... More
Connection probabilities and RSW-type bounds for the FK Ising modelDec 21 2009We prove Russo-Seymour-Welsh-type uniform bounds on crossing probabilities for the FK Ising model at criticality, independent of the boundary conditions. Our proof relies mainly on Smirnov's fermionic observable for the FK Ising model, which allows us ... More
A quantitative Burton-Keane estimate under strong FKG conditionSep 18 2014Sep 22 2016We consider translationally-invariant percolation models on $\mathbb{Z}^d$ satisfying the finite energy and the FKG properties. We provide explicit upper bounds on the probability of having two distinct clusters going from the endpoints of an edge to ... More
Absence of infinite cluster for critical Bernoulli percolation on slabsJan 28 2014We prove that for Bernoulli percolation on a graph $\mathbb{Z}^2\times\{0,\dots,k\}$ ($k\ge 0$), there is no infinite cluster at criticality, almost surely. The proof extends to finite range Bernoulli percolation models on $\mathbb{Z}^2$ which are invariant ... More
Supercritical self-avoiding walks are space-fillingOct 13 2011Sep 25 2012We consider random self-avoiding walks between two points on the boundary of a finite subdomain of Z^d (the probability of a self-avoiding trajectory gamma is proportional to mu^{-length(gamma)}). We show that the random trajectory becomes space-filling ... More
On the critical parameters of the $q\ge4$ random-cluster model on isoradial graphsJul 06 2015The critical surface for random-cluster model with cluster-weight $q\ge 4$ on isoradial graphs is identified using parafermionic observables. Correlations are also shown to decay exponentially fast in the subcritical regime. While this result is restricted ... More
A new computation of the critical point for the planar random-cluster model with $q\ge1$Apr 13 2016We present a new computation of the critical value of the random-cluster model with cluster weight $q\ge 1$ on $\mathbb{Z}^2$. This provides an alternative approach to the result of Beffara and Duminil-Copin. We believe that this approach has several ... More
Crossing probabilities in topological rectangles for the critical planar FK-Ising modelDec 30 2013We consider the FK-Ising model in two dimensions at criticality. We obtain bounds on crossing probabilities of arbitrary topological rectangles, uniform with respect to the boundary conditions, generalizing results of [DCHN11] and [CS12]. Our result relies ... More
Continuity of the phase transition for planar random-cluster and Potts models with $1\le q\le4$May 15 2015This article studies the planar Potts model and its random-cluster representation. We show that the phase transition of the nearest-neighbor ferromagnetic $q$-state Potts model on $\mathbb Z^2$ is continuous for $q\in\{2,3,4\}$, in the sense that there ... More
Exponential decay of truncated correlations for the Ising model in any dimension for all but the critical temperatureAug 01 2018The truncated two-point function of the ferromagnetic Ising model on $\mathbb Z^d$ ($d\ge3$) in its pure phases is proven to decay exponentially fast throughout the ordered regime ($\beta>\beta_c$ and $h=0$). Together with the previously known results, ... More
Conformal invariance of crossing probabilities for the Ising model with free boundary conditionsOct 14 2014May 10 2016We prove that crossing probabilities for the critical planar Ising model with free boundary conditions are conformally invariant in the scaling limit, a phenomenon first investigated numerically by Langlands, Lewis and Saint-Aubin. We do so by establishing ... More
Sharp phase transition for the random-cluster and Potts models via decision treesMay 08 2017Dec 23 2018We 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
Subcritical phase of $d$-dimensional Poisson-Boolean percolation and its vacant setMay 02 2018Nov 04 2018We prove that the Poisson-Boolean percolation on $\mathbb{R}^d$ undergoes a sharp phase transition in any dimension under the assumption that the radius distribution has a $5d-3$ finite moment (in particular we do not assume that the distribution is bounded). ... More
Exponential decay of connection probabilities for subcritical Voronoi percolation in $\mathbb{R}^d$May 22 2017We prove that for Voronoi percolation on $\mathbb{R}^d$, there exists $p_c\in[0,1]$ such that - for $p<p_c$, there exists $c_p>0$ such that $\mathbb{P}_p[0\text{ connected to distance }n]\leq \exp(-c_p n)$, - there exists $c>0$ such that for $p>p_c$, ... More
On the Gibbs states of the noncritical Potts model on Z^2May 21 2012Feb 15 2013We prove that all Gibbs states of the q-state nearest neighbor Potts model on Z^2 below the critical temperature are convex combinations of the q pure phases; in particular, they are all translation invariant. To achieve this goal, we consider such models ... More
Bounding the number of self-avoiding walks: Hammersley-Welsh with polygon insertionSep 04 2018Let $c_n = c_n(d)$ denote the number of self-avoiding walks of length $n$ starting at the origin in the Euclidean nearest-neighbour lattice $\mathbb{Z}^d$. Let $\mu = \lim_n c_n^{1/n}$ denote the connective constant of $\mathbb{Z}^d$. In 1962, Hammersley ... More
Brochette percolationAug 17 2016Apr 20 2017We study bond percolation on the square lattice with one-dimensional inhomogeneities. Inhomogeneities are introduced in the following way: A vertical column on the square lattice is the set of vertical edges that project to the same vertex on $\mathbb{Z}$. ... More
Universality of two-dimensional critical cellular automataJun 25 2014Mar 13 2017We study the class of monotone, two-state, deterministic cellular automata, in which sites are activated (or `infected') by certain configurations of nearby infected sites. These models have close connections to statistical physics, and several specific ... More
On the probability that self-avoiding walk ends at a given pointMay 06 2013May 01 2014We prove two results on the delocalization of the endpoint of a uniform self-avoiding walk on Z^d for d>1. We show that the probability that a walk of length n ends at a point x tends to 0 as n tends to infinity, uniformly in x. Also, for any fixed x ... More
Universality of two-dimensional critical cellular automataJun 25 2014Jul 22 2015We study the class of monotone, two-state, deterministic cellular automata, in which sites are activated (or `infected') by certain configurations of nearby infected sites. These models have close connections to statistical physics, and several specific ... More
Higher order corrections for anisotropic bootstrap percolationNov 10 2016Oct 09 2017We study the critical probability for the metastable phase transition of the two-dimensional anisotropic bootstrap percolation model with $(1,2)$-neighbourhood and threshold $r = 3$. The first order asymptotics for the critical probability were recently ... More
Brochette percolationAug 17 2016We study bond percolation on the square lattice with one-dimensional inhomogeneities. Inhomogeneities are introduced in the following way: A vertical column on the square lattice is the set of vertical edges that project to the same vertex on $\mathbb{Z}$. ... More
Disorder, entropy and harmonic functionsNov 21 2011Oct 28 2015We study harmonic functions on random environments with particular emphasis on the case of the infinite cluster of supercritical percolation on $\mathbb{Z}^d$. We prove that the vector space of harmonic functions growing at most linearly is $(d+1)$-dimensional ... More
Lectures on Self-Avoiding WalksJun 11 2012These lecture notes provide a rapid introduction to a number of rigorous results on self-avoiding walks, with emphasis on the critical behaviour. Following an introductory overview of the central problems, an account is given of the Hammersley--Welsh ... More
Higher order corrections for anisotropic bootstrap percolationNov 10 2016We study the critical probability for the metastable phase transition of the two-dimensional anisotropic bootstrap percolation model with $(1,2)$-neighbourhood and threshold $r = 3$. The first order asymptotics for the critical probability were recently ... More
Containing Internal Diffusion Limited AggregationNov 02 2011Internal Diffusion Limited Aggregation (IDLA) is a model that describes the growth of a random aggregate of particles from the inside out. Shellef proved that IDLA processes on supercritical percolation clusters of integer-lattices fill Euclidean balls, ... More
Internal Diffusion-Limited aggregation with uniform starting pointsJul 11 2017We study internal diffusion-limited aggregation with random starting points on Z^d. In this model, each new particle starts from a vertex chosen uniformly at random on the existing aggregate. We prove that the limiting shape of the aggregate is a Euclidean ... More
The sharp threshold for the Duarte modelMar 16 2016Oct 10 2016The class of critical bootstrap percolation models in two dimensions was recently introduced by Bollob\'as, Smith and Uzzell, and the critical threshold for percolation was determined up to a constant factor for all such models by the authors of this ... More
Emergent Planarity in two-dimensional Ising Models with finite-range InteractionsJan 15 2018Mar 05 2018The known Pfaffian structure of the boundary spin correlations, and more generally order-disorder correlation functions, is given a new explanation through simple topological considerations within the model's random current representation. This perspective ... More
The sharp threshold for bootstrap percolation in all dimensionsOct 16 2010Feb 24 2011In r-neighbour bootstrap percolation on a graph G, a (typically random) set A of initially 'infected' vertices spreads by infecting (at each time step) vertices with at least r already-infected neighbours. This process may be viewed as a monotone version ... More
Minimal growth harmonic functions on lamplighter groupsJul 04 2016We study the minimal possible growth of harmonic functions on lamplighters. We find that $(\mathbb{Z}/2)\wr \mathbb{Z}$ has no sublinear harmonic functions, $(\mathbb{Z}/2)\wr \mathbb{Z}^2$ has no sublogarithmic harmonic functions, and neither has the ... More
Seven-dimensional forest firesFeb 27 2013Jul 03 2015We show that in high dimensional Bernoulli percolation, removing from a thin infinite cluster a much thinner infinite cluster leaves an infinite component. This observation has implications for the van den Berg-Brouwer forest fire process, also known ... More
Universality for the random-cluster model on isoradial graphsNov 07 2017We show that the canonical random-cluster measure associated to isoradial graphs is critical for all $q \geq 1$. Additionally, we prove that the phase transition of the model is of the same type on all isoradial graphs: continuous for $1 \leq q \leq 4$ ... More
Law of the Iterated Logarithm for the random walk on the infinite percolation clusterSep 25 2008We show that random walks on the infinite supercritical percolation clusters in Z^d satisfy the usual Law of the Iterated Logarithm. The proof combines Barlow's Gaussian heat kernel estimates and the ergodicity of the random walk on the environment viewed ... More
The critical fugacity for surface adsorption of self-avoiding walks on the honeycomb lattice is $1+\sqrt{2}$Sep 02 2011Sep 24 2013In 2010, Duminil-Copin and Smirnov proved a long-standing conjecture of Nienhuis, made in 1982, that the growth constant of self-avoiding walks on the hexagonal (a.k.a. honeycomb) lattice is $\mu=\sqrt{2+\sqrt{2}}.$ A key identity used in that proof was ... More
Discontinuity of the phase transition for the planar random-cluster and Potts models with $q>4$Nov 29 2016Sep 05 2017We prove that the $q$-state Potts model and the random-cluster model with cluster weight $q>4$ undergo a discontinuous phase transition on the square lattice. More precisely, we show - Existence of multiple infinite-volume measures for the critical Potts ... More
Discontinuity of the phase transition for the planar random-cluster and Potts models with $q>4$Nov 29 2016We prove that the $q$-state Potts model and the random-cluster model with cluster weight $q>4$ undergo a discontinuous phase transition on the square lattice. More precisely, we show - Existence of multiple infinite-volume measures for the critical Potts ... More
Convergence of Ising interfaces to Schramm's SLE curvesDec 02 2013Dec 31 2013We show how to combine our earlier results to deduce strong convergence of the interfaces in the planar critical Ising model and its random-cluster representation to Schramm's SLE curves with parameter $\kappa=3$ and $\kappa=16/3$ respectively.
On the number of maximal paths in directed last-passage percolationJan 17 2018We show that the number of maximal paths in directed last-passage percolation on the hypercubic lattice ${\mathbb Z}^d$ $(d\geq2)$ in which weights take finitely many values is typically exponentially large.
Adaptive Spatial Binning of 2D Spectra and Images Using Voronoi TessellationsFeb 20 2002We present new techniques to perform adaptive spatial binning of two-dimensional (2D) data to reach a chosen constant signal-to-noise ratio per bin. These methods are required particularly for the proper analysis of Integral Field Spectrograph (IFS) observations, ... More
Axisymmetric dynamical models for SAURON and OASIS observations of NGC 3377Nov 17 2003Nov 18 2003We present a unique set of nested stellar kinematical maps of NGC 3377 obtained with the integral-field spectrographs OASIS and SAURON. We then construct general axisymmetric dynamical models for this galaxy, based on the Schwarzschild numerical orbit ... More
A basic introduction to large deviations: Theory, applications, simulationsJun 21 2011Feb 29 2012The theory of large deviations deals with the probabilities of rare events (or fluctuations) that are exponentially small as a function of some parameter, e.g., the number of random components of a system, the time over which a stochastic system is observed, ... More
Simple spin models with non-concave entropiesApr 01 2005Jan 24 2008Two simple spin models are studied to show that the microcanonical entropy can be a non-concave function of the energy, and that the microcanonical and canonical ensembles can give non-equivalent descriptions of the same system in the thermodynamic limit. ... More
Repeated Binomial Coefficients and High-Degree CurvesNov 15 2014We consider the problem of characterizing solutions in $(x, y)$ to the equation ${x \choose y}={{x-a} \choose {y+b}}$ in terms of $a$ and $b$. We obtain one simple result which allows the determination of a ratio in terms of $a$ and $b$ which the ratio ... More
The homology systole of hyperbolic Riemann surfacesOct 02 2010Apr 07 2011The main goal of this note is to show that the study of closed hyperbolic surfaces with maximum length systole is in fact the study of surfaces with maximum length homological systole. The same result is shown to be true for once-punctured surfaces, and ... More
Methods for calculating nonconcave entropiesMar 01 2010Apr 28 2010Five different methods which can be used to analytically calculate entropies that are nonconcave as functions of the energy in the thermodynamic limit are discussed and compared. The five methods are based on the following ideas and techniques: i) microcanonical ... More
Comment on "First-order phase transitions: equivalence between bimodalities and the Yang-Lee theorem"Mar 02 2005I discuss the validity of a result put forward recently by Chomaz and Gulminelli [Physica A 330 (2003) 451] concerning the equivalence of two definitions of first-order phase transitions. I show that distributions of zeros of the partition function fulfilling ... More
Fixed point free involutions on Riemann surfacesApr 06 2005Jul 09 2005Involutions without fixed points on hyperbolic closed Riemann surface are discussed. For an orientable surface $X$ of even genus with an arbitrary Riemannian metric $d$ admitting an involution $\tau$, it is known that $\min_{p\in X}d(p,\tau(p))$ is bounded ... More
Annealed scaling relations for Voronoi percolationJun 21 2018Sep 28 2018We prove annealed scaling relations for planar Voronoi percolation. To our knowledge, this is the first result of this kind for a continuum percolation model. We are mostly inspired by the proof of scaling relations for Bernoulli percolation by Kesten ... More
The Monadic Second Order Theory of Grid-Free 1-Safe Petri Nets is DecidableFeb 09 2018Feb 16 2018Finite 1-safe Petri nets, also called \emph{net systems}, are natural models of asynchronous concurrency. The event structure of a net system describes all its possible executions and their concurrent nature: two events may be causally ordered, occur ... More
A short note on short pantsApr 28 2013May 22 2013It is a theorem of Bers that any closed hyperbolic surface admits a pants decomposition consisting of curves of bounded length where the bound only depends on the topology of the surface. The question of the quantification of the optimal constants has ... More
A Note on the Forward-Douglas--Rachford Splitting for Monotone Inclusion and Convex OptimizationApr 23 2017Apr 27 2018We shed light on the structure of the "three-operator" version of the forward-Douglas--Rachford splitting algorithm for finding a zero of a sum of maximally monotone operators $A + B + C$, where $B$ is cocoercive, involving only the computation of $B$ ... More
The large deviation approach to statistical mechanicsApr 02 2008Aug 20 2009The theory of large deviations is concerned with the exponential decay of probabilities of large fluctuations in random systems. These probabilities are important in many fields of study, including statistics, finance, and engineering, as they often yield ... More
A Class of Zielonka Automata with a Decidable Controller Synthesis ProblemJan 20 2016Feb 02 2016The decidability of the distributed version of the Ramadge and Wonham control problem (Ramadge and Wonham 1989), where both the plant and the controllers are modelled as Zielonka au-tomata (Zielonka 1987; Diekert and Rozenberg 1995) is a challenging open ... More
Relationships between p-unit constructions for real quadratic fieldsApr 10 2010Let $K$ be a real quadratic field and let $p$ be a prime number which is inert in $K$. Let $K_p$ be the completion of $K$ at $p$. In a previous paper, we constructed a $p$-adic invariant $u_C\in K_p$, and we proved a $p$-adic Kronecker limit formula relating ... More
Ensemble equivalence for general many-body systemsJun 15 2011Nov 15 2011It has been proved for a class of mean-field and long-range systems that the concavity of the thermodynamic entropy determines whether the microcanonical and canonical ensembles are equivalent at the level of their equilibrium states, i.e., whether they ... More
Temperature fluctuations and mixtures of equilibrium states in the canonical ensembleDec 12 2002Dec 12 2002It has been suggested recently that `$q$-exponential' distributions which form the basis of Tsallis' non-extensive thermostatistical formalism may be viewed as mixtures of exponential (Gibbs) distributions characterized by a fluctuating inverse temperature. ... More
When is a quantity additive, and when is it extensive?Jan 09 2002The difference between the terms additivity and extensivity, as well as their respective negations, is critically analyzed and illustrated with a few examples. The concepts of subadditivity, pseudo-additivity, and pseudo-extensivity are also defined.
Comment on "Entropy Generation in Computation and the Second law of Thermodynamics", by S. Ishioka and N. FuchikamiFeb 19 1999This brief note argues that, contrary to the claim of Ishioka and Fuchikami (chao-dyn/9902012), Landauer's principle is concerned a priori with entropy generation in computing processes. The concept of heat, in this principle, is only relevant when a ... More
Multiplicities of simple closed geodesics and hypersurfaces in Teichmüller spaceJan 29 2007Jul 25 2007Using geodesic length functions, we define a natural family of real codimension 1 subvarieties of Teichm\"uller space, namely the subsets where the lengths of two distinct simple closed geodesics are of equal length. We investigate the point set topology ... More
Propagator Dyson-Schwinger Equations of Coulomb Gauge Yang-Mills Theory Within the First Order FormalismDec 12 2006Coulomb gauge Yang-Mills theory within the first order formalism is considered with a view to deriving the propagator Dyson-Schwinger equations. The first order formalism is studied with special emphasis on the BRS invariance and it is found that there ... More
D-branes in Orbifold Singularities and Equivariant K-TheoryDec 23 1998May 19 1999The study of brane-antibrane configurations in string theory leads to the understanding of supersymmetric D$p$-branes as the bound states of higher dimensional branes. Configurations of pairs brane-antibrane do admit in a natural way their description ... More
Chiral condensates and QCD vacuum in two dimensionsApr 02 1997We analyze the chiral symmetries of flavored quantum chromodynamics in two dimensions and show the existence of chiral condensates within the path-integral approach. The massless and massive cases are discussed as well, for arbitrary finite and infinite ... More
Gravitational Lensing of Distant SupernovaeMar 21 2005We use a series of ray-tracing experiments to determine the magnification distribution of high-redshift sources by gravitational lensing. We determine empirically the relation between magnification and redshift, for various cosmological models. We then ... More
Perturbation Theory of Coulomb Gauge Yang-Mills Theory Within the First Order FormalismSep 03 2007Perturbative Coulomb gauge Yang-Mills theory within the first order formalism is considered. Using a differential equation technique and dimensional regularization, analytic results for both the ultraviolet divergent and finite parts of the two-point ... More
N=2 String Geometry and the Heavenly EquationsMay 22 2004Mar 22 2006In this paper we survey some of the relations between Plebanski description of self-dual gravity through the heavenly equations and the physics (and mathematics) of N=2 Strings. In particular we focus on the correspondence between the infinite hierarchy ... More
Construction of hyperbolic Riemann surfaces with large systolesMay 23 2013Aug 15 2016Let $S$ be a compact hyperbolic Riemann surface of genus $g \geq 2$. We call a systole a shortest simple closed geodesic in $S$ and denote by $\mathop{sys}(S)$ its length. Let $\mathop{msys(g)}$ be the maximal value that $\mathop{sys}(\cdot)$ can attain ... More
On a generalization of the logistic mapMar 07 2000A family of non-conjugate chaotic maps generalizing the well-known logistic function is defined, and some of its basic properties studied. A simple formula for the Lyapunov exponents of all the maps contained in this family is given based on the construction ... More