arXiv subjects

Charles M. Newman

Publications and source records attributed to Charles M. Newman.

At least 37 records · Page 2Linked to original sources

arXiv2017

Zero-temperature dynamics in the dilute Curie-Weiss model

We consider the Ising model on a dense Erdős--Rényi random graph, $\mathcal G(N,p)$, with $p>0$ fixed---equivalently, a disordered Curie--Weiss Ising model with $\mbox{Ber}(p)$ couplings---at zero temperature. The disorder may induce local energy minima in addition to the two uniform ground states. In this paper we prove that, starting from a typical initial configuration, the zero-temperature dynamics avoids all such local minima and absorbs into a predetermined one of the two uniform ground states. We relate this to the local MINCUT problem on dense random graphs; namely with high probability, the greedy search for a local MINCUT of $\mathcal G(N,p)$ with $p>0$ fixed, started from a uniform random partition, fails to find a non-trivial cut. In contrast, in the disordered Curie--Weiss model with heavy-tailed couplings, we demonstrate that zero-temperature dynamics has positive probability of absorbing in a random local minimum different from the two homogenous ground states.

math.PR↗

arXiv2016

Gaussian fluctuations for the classical XY model

We study the classical XY model in bounded domains of $\mathbb{Z}^{d}$ with Dirichlet boundary conditions. We prove that when the temperature goes to zero faster than a certain rate as the lattice spacing goes to zero, the fluctuation field converges to a standard Gaussian white noise. This and related results also apply to a large class of gradient field models.

math.PR↗

arXiv2015

Critical Percolation and the Minimal Spanning Tree in Slabs

The minimal spanning forest on $\mathbb{Z}^{d}$ is known to consist of a single tree for $d \leq 2$ and is conjectured to consist of infinitely many trees for large $d$. In this paper, we prove that there is a single tree for quasi-planar graphs such as $\mathbb{Z}^{2}\times {\{0,\ldots,k\}}^{d-2}$. Our method relies on generalizations of the "Gluing Lemma" of arXiv:1401.7130. A related result is that critical Bernoulli percolation on a slab satisfies the box-crossing property. Its proof is based on a new Russo-Seymour-Welsh type theorem for quasi-planar graphs. Thus, at criticality, the probability of an open path from $0$ of diameter $n$ decays polynomially in $n$. This strengthens the result of arXiv:1401.7130, where the absence of an infinite cluster at criticality was first established.

math.PR↗

arXiv2015

Fixation to Consensus on Tree-related Graphs

We study a continuous time Markov process whose state space consists of an assignment of +1 or -1 to each vertex of a graph G. The graphs that we treat are related to homogeneous trees of degree K $\geq$ 3, such as finite or infinite stacks of such trees. The initial spin configuration is chosen from a Bernoulli product measure with density $θ$ of +1 spins. The system evolves according to an agreement inducing dynamics: each vertex, at rate 1, changes its spin value to agree with the majority of its neighbors. We study the long time behavior of this system and prove that, if $θ$ is close enough to 1, the system reaches fixation to consensus. The geometric percolation-type arguments introduced here may be of independent interest.

math.PR↗

arXiv2015

Planar Ising magnetization field I. Uniqueness of the critical scaling limit

The aim of this paper is to prove the following result. Consider the critical Ising model on the rescaled grid $a\mathbb{Z}^2$, then the renormalized magnetization field \[Φ^a:=a^{15/8}\sum_{x\in a\mathbb{Z}^2}σ_xδ_x,\] seen as a random distribution (i.e., generalized function) on the plane, has a unique scaling limit as the mesh size $a\searrow0$. The limiting field is conformally covariant.

math.PR↗

arXiv2014

Coarsening dynamics on $\mathbb{Z}^d$ with frozen vertices

We study Markov processes in which $\pm 1$-valued random variables $σ_x(t), x\in \mathbb{Z}^d$, update by taking the value of a majority of their nearest neighbors or else tossing a fair coin in case of a tie. In the presence of a random environment of frozen plus (resp., minus) vertices with density $ρ^+$ (resp., $ρ^-$), we study the prevalence of vertices that are (eventually) fixed plus or fixed minus or flippers (changing forever). Our main results are that, for $ρ^+ >0$ and $ρ^- =0$, all sites are fixed plus, while for $ρ^+ >0$ and $ρ^-$ very small (compared to $ρ^+$), the fixed minus and flippers together do not percolate. We also obtain some results for deterministic placement of frozen vertices.

math.PR↗

arXiv2014

The voter model chordal interface in two dimensions

Consider the voter model on a box of side length $L$ (in the triangular lattice) with boundary votes fixed forever as type 0 or type 1 on two different halves of the boundary. Motivated by analogous questions in percolation, we study several geometric objects at stationarity, as $L\rightarrow \infty$. One is the interface between the (large -- i.e., boundary connected) 0-cluster and 1-cluster. Another is the set of large "coalescing classes" determined by the coalescing walk process dual to the voter model.

math.PR↗

arXiv2014

Fluctuation Bounds For Interface Free Energies in Spin Glasses

We consider the free energy difference restricted to a finite volume for certain pairs of incongruent thermodynamic states (if they exist) in the Edwards-Anderson Ising spin glass at nonzero temperature. We prove that the variance of this quantity with respect to the couplings grows proportionally to the volume in any dimension greater than or equal to two. As an illustration of potential applications, we use this result to restrict the possible structure of Gibbs states in two dimensions.

math-ph↗

arXiv2013

Fixation for coarsening dynamics in 2D slabs

For the zero temperature limit of Ising Glauber Dynamics on 2D slabs the existence or nonexistence of vertices that do not fixate is determined as a function of slab thickness.

math.PR↗

arXiv2013

The Brownian Net with Killing

Motivated by its relevance for the study of perturbations of one-dimensional voter models, including stochastic Potts models at low temperature, we consider diffusively rescaled coalescing random walks with branching and killing. Our main result is convergence to a new continuum process, in which the random space-time paths of the Sun-Swart Brownian net are terminated at a Poisson cloud of killing points. We also prove existence of a percolation transition as the killing rate varies. Key issues for convergence are the relations of the discrete model killing points and their Poisson intensity measure to the continuum counterparts.

math.PR↗

arXiv2013

Planar Ising magnetization field II. Properties of the critical and near-critical scaling limits

In [CGN12], we proved that the renormalized critical Ising magnetization fields $Φ^a:= a^{15/8} \sum_{x\in a\, \Z^2} σ_x \, δ_x$ converge as $a\to 0$ to a random distribution that we denoted by $Φ^\infty$. The purpose of this paper is to establish some fundamental properties satisfied by this $Φ^\infty$ and the near-critical fields $Φ^{\infty,h}$. More precisely, we obtain the following results. \bi [(i)] If $A\subset \C$ is a smooth bounded domain and if $m=m_A := <{Φ^\infty, 1_A}$ denotes the limiting rescaled magnetization in $A$, then there is a constant $c=c_A>0$ such that {equation*} \log \Pb{m > x} \underset{x\to \infty}{\sim} -c \; x^{16}\,.{equation*} In particular, this provides an alternative proof that the field $Φ^\infty$ is non-Gaussian (another proof of this fact would use the $n$-point correlation functions established in \cite{CHI} which do not satisfy Wick's formula). [(ii)] The random variable $m=m_A$ has a smooth {\it density} and one has more precisely the following bound on its Fourier transform: $|\Eb{e^{i\,t m}} |\le e^{- \tilde{c}\, |t|^{16/15}}$. [(iii)] There exists a one-parameter family $Φ^{\infty,h}$ of near-critical scaling limits for the magnetization field in the plane with vanishingly small external magnetic field. \ei

math.PR↗

arXiv2013

The Ising magnetization exponent on Z^2 is 1/15

We prove that for the Ising model defined on the plane $\Z^2$ at $β=β_c$, the average magnetization under an external magnetic field $h>0$ behaves exactly like \[{σ_0}_{β_c, h} \asymp h^{\frac 1 {15}}\,. \] The proof, which is surprisingly simple compared to an analogous result for percolation (i.e. that $θ(p)=(p-p_c)^{5/36+o(1)}$ on the triangular lattice \cite{\SmirnovWerner,\KestenScaling}) relies on the GHS inequality as well as the RSW theorem for FK percolation from \cite{\RSWfk}. The use of GHS to obtain inequalities involving critical exponents is not new; in this paper we show how it can be combined with RSW to obtain matching upper and lower bounds for the average magnetization.

math.PR↗

arXiv2013

Coarsening in 2D slabs

We study coarsening; that is, the zero-temperature limit of Glauber dynamics in the standard Ising model on slabs S_k = Z^2 x {0, ..., k-1} of all thicknesses k \geq 2 (with free and periodic boundary conditions in the third coordinate). We show that with free boundary conditions, for k \geq 3, some sites fixate for large times and some do not, whereas for k=2, all sites fixate. With periodic boundary conditions, for k \geq 4, some sites fixate and others do not, while for k=2 and 3, all sites fixate.

math.PR↗

arXiv2012

Absence of site percolation at criticality in Z^2 x {0,1}

In this note we consider site percolation on a two dimensional sandwich of thickness two, the graph Z^2 x {0,1}. We prove that there is no percolation at the critical point. The same arguments are valid for a sandwich of thickness three with periodic boundary conditions. It remains an open problem to extend this result to other sandwiches.

math.PR↗

arXiv2008

Ising (Conformal) Fields and Cluster Area Measures

We provide a representation for the scaling limit of the d=2 critical Ising magnetization field as a (conformal) random field using SLE (Schramm-Loewner Evolution) clusters and associated renormalized area measures. The renormalized areas are from the scaling limit of the critical FK (Fortuin-Kasteleyn) clusters and the random field is a convergent sum of the area measures with random signs. Extensions to off-critical scaling limits, to d=3 and to Potts models are also considered.

math.PR↗

arXiv2006

Critical Percolation Exploration Path and SLE(6): a Proof of Convergence

It was argued by Schramm and Smirnov that the critical site percolation exploration path on the triangular lattice converges in distribution to the trace of chordal SLE(6). We provide here a detailed proof, which relies on Smirnov's theorem that crossing probabilities have a conformally invariant scaling limit (given by Cardy's formula). The version of convergence to SLE(6) that we prove suffices for the Smirnov-Werner derivation of certain critical percolation crossing exponents and for our analysis of the critical percolation full scaling limit as a process of continuum nonsimple loops.

math.PR↗

arXiv2006

SLE(6) and CLE(6) from Critical Percolation

We review some of the recent progress on the scaling limit of two-dimensional critical percolation; in particular, the convergence of the exploration path to chordal SLE(6) and the "full" scaling limit of cluster interface loops. The results given here on the full scaling limit and its conformal invariance extend those presented previously. For site percolation on the triangular lattice, the results are fully rigorous. We explain some of the main ideas, skipping most technical details.

math.PR↗

arXiv2006

Two-Dimensional Critical Percolation: The Full Scaling Limit

We use SLE(6) paths to construct a process of continuum nonsimple loops in the plane and prove that this process coincides with the full continuum scaling limit of 2D critical site percolation on the triangular lattice -- that is, the scaling limit of the set of all interfaces between different clusters. Some properties of the loop process, including conformal invariance, are also proved.

math.PR↗