arXiv subjects

Charles M. Newman

Publications and source records attributed to Charles M. Newman.

At least 19 recordsLinked to original sources

arXiv2025

Zero Temperature Dynamics of Ising Systems on Hypercubes

We study the zero-temperature Glauber dynamics of homogeneous Ising ferromagnets on hypercubes, as their dimension d varies. We investigate the asymptotic (d goes to infinity and time t goes to infinity) behavior of various quantities on hypercubes, such as the final magnetization, the probability for the system to enter a ground state, etc. Our numerical studies are carried out using a uniformly random initial state but with the constraint that the initial magnetization is zero. The final states can be divided into three categories: ground states, frozen states, and blinker states. We use the notion of a k-core to describe the geometry of the frozen states and give an exponential lower bound for the number of frozen states in terms of d. Blinker states -- which exist only in even d -- are final states containing at least one blinker (a permanently flipping spin). Blinker states can have rich local structures; we give explicit constructions for configurations that contain blinkers and prove that the lowest possible dimension for blinker configurations is d = 8. We also study the 'Nature vs. Nurture' problem on hypercubes, asking how much the final state depends on the information contained in the initial configuration, and how much depends on the realization of the dynamical evolution. Finally, we provide several conjectures and suggest some open problems based on the numerical results.

cond-mat.stat-mech↗

arXiv2023

Monotonicity of Ursell functions in the Ising model

In this paper, we consider Ising models with ferromagnetic pair interactions. We prove that the Ursell functions $u_{2k}$ satisfy: $(-1)^{k-1}u_{2k}$ is increasing in each interaction. As an application, we prove a 1983 conjecture by Nishimori and Griffiths about the partition function of the Ising model with complex external field $h$: its closest zero to the origin (in the variable $h$) moves towards the origin as an arbitrary interaction increases.

math.PR↗

arXiv2022

Thermodynamic limit of the first Lee-Yang zero

We complete the verification of the 1952 Yang and Lee proposal that thermodynamic singularities are exactly the limits in ${\mathbb R}$ of finite-volume singularities in ${\mathbb C}$. For the Ising model defined on a finite $Λ\subset\mathbb{Z}^d$ at inverse temperature $β\geq0$ and external field $h$, let $α_1(Λ,β)$ be the modulus of the first zero (that closest to the origin) of its partition function (in the variable $h$). We prove that $α_1(Λ,β)$ decreases to $α_1(\mathbb{Z}^d,β)$ as $Λ$ increases to $\mathbb{Z}^d$ where $α_1(\mathbb{Z}^d,β)\in[0,\infty)$ is the radius of the largest disk centered at the origin in which the free energy in the thermodynamic limit is analytic. We also note that $α_1(\mathbb{Z}^d,β)$ is strictly positive if and only if $β$ is strictly less than the critical inverse temperature.

math.PR↗

arXiv2022

Motion of Lee-Yang zeros

We consider the zeros of the partition function of the Ising model with ferromagnetic pair interactions and complex external field. Under the assumption that the graph with strictly positive interactions is connected, we vary the interaction (denoted by $t$) at a fixed edge. It is already known that each zero is monotonic (either increasing or decreasing) in $t$; we prove that its motion is local: the entire trajectories of any two distinct zeros are disjoint. If the underlying graph is a complete graph and all interactions take the same value $t\geq 0$ (i.e., the Curie-Weiss model), we prove that all the principal zeros (those in $i[0,π/2)$) decrease strictly in $t$.

math-ph↗

arXiv2022

Ising model with Curie-Weiss perturbation

Consider the nearest-neighbor Ising model on $Λ_n:=[-n,n]^d\cap\mathbb{Z}^d$ at inverse temperature $β\geq 0$ with free boundary conditions, and let $Y_n(σ):=\sum_{u\inΛ_n}σ_u$ be its total magnetization. Let $X_n$ be the total magnetization perturbed by a critical Curie-Weiss interaction, i.e., \begin{equation*} \frac{d F_{X_n}}{d F_{Y_n}}(x):=\frac{\exp[x^2/\left(2\langle Y_n^2 \rangle_{Λ_n,β}\right)]}{\left\langle\exp[Y_n^2/\left(2\langle Y_n^2\rangle_{Λ_n,β}\right)]\right\rangle_{Λ_n,β}}, \end{equation*} where $F_{X_n}$ and $F_{Y_n}$ are the distribution functions for $X_n$ and $Y_n$ respectively. We prove that for any $d\geq 4$ and $β\in[0,β_c(d)]$ where $β_c(d)$ is the critical inverse temperature, any subsequential limit (in distribution) of $\{X_n/\sqrt{\mathbb{E}\left(X_n^2\right)}:n\in\mathbb{N}\}$ has an analytic density (say, $f_X$) all of whose zeros are pure imaginary, and $f_X$ has an explicit expression in terms of the asymptotic behavior of zeros for the moment generating function of $Y_n$. We also prove that for any $d\geq 1$ and then for $β$ small, \begin{equation*} f_X(x)=K\exp(-C^4x^4), \end{equation*} where $C=\sqrt{Γ(3/4)/Γ(1/4)}$ and $K=\sqrt{Γ(3/4)}/(4Γ(5/4)^{3/2})$. Possible connections between $f_X$ and the high-dimensional critical Ising model with periodic boundary conditions are discussed.

math.PR↗

arXiv2021

The Gaussian process for particle masses in the near-critical Ising model

We review the construction of a stationary Gaussian process $X(t)$ starting from the near-critical continuum scaling limit $Φ^h$ of the Ising magnetization and its relation to the mass spectrum of the relativistic quantum field theory associated to $Φ^h$. Then for the near-critical Ising model on $a \mathbb{Z}^2$ with external field $a^{15/8} h$, we study the renormalized magnetization along a vertical line (with horizontal coordinate approximately $t$) and prove that the limit as $a\downarrow 0$ is the same Gaussian process $X(t)$. We also explore the possible extension of this approach to dimensions $d > 2$.

math.PR↗

arXiv2020

Conformal Measure Ensembles and Planar Ising Magnetization: A Review

We provide a review of results on the critical and near-critical scaling limit of the planar Ising magnetization field obtained in the past dozen years. The results are presented in the framework of coupled loop and measure ensembles, and some new proofs are provided.

math.PR↗

arXiv2020

The effect of free boundary conditions on the Ising model in high dimensions

We study the critical Ising model with free boundary conditions on finite domains in $\mathbb{Z}^d$ with $d\geq4$. Under the assumption, so far only proved completely for high $d$, that the critical infinite volume two-point function is of order $|x-y|^{-(d-2)}$ for large $|x-y|$, we prove the same is valid on large finite cubes with free boundary conditions, as long as $x, y$ are not too close to the boundary. This confirms a numerical prediction in the physics literature by showing that the critical susceptibility in a finite domain of linear size $L$ with free boundary conditions is of order $L^2$ as $L\rightarrow\infty$. We also prove that the scaling limit of the near-critical (small external field) Ising magnetization field with free boundary conditions is Gaussian with the same covariance as the critical scaling limit, and thus the correlations do not decay exponentially. This is very different from the situation in low $d$ or the expected behavior in high $d$ with bulk boundary conditions.

math.PR↗

arXiv2019

A Gaussian process related to the mass spectrum of the near-critical Ising model

Let $Φ^h(x)$ with $x=(t,y)$ denote the near-critical scaling limit of the planar Ising magnetization field. We take the limit of $Φ^h$ as the spatial coordinate $y$ scales to infinity with $t$ fixed and prove that it is a stationary Gaussian process $X(t)$ whose covariance function is the Laplace transform of a mass spectral measure $ρ$ of the relativistic quantum field theory associated to the Euclidean field $Φ^h$. Our analysis of the small distance/time behavior of the covariance functions of $Φ^h$ and $X(t)$ shows that $ρ$ is finite but has infinite first moment.

math.PR↗

arXiv2019

Local minima in disordered mean-field ferromagnets

We consider the complexity of random ferromagnetic landscapes on the hypercube $\{\pm 1\}^N$ given by Ising models on the complete graph with i.i.d. non-negative edge-weights. This includes, in particular, the case of Bernoulli disorder corresponding to the Ising model on a dense random graph $\mathcal G(N,p)$. Previous results had shown that, with high probability as $N\to\infty$, the gradient search (energy-lowering) algorithm, initialized uniformly at random, converges to one of the homogeneous global minima (all-plus or all-minus). Here, we devise two modified algorithms tailored to explore the landscape at near-zero magnetizations (where the effect of the ferromagnetic drift is minimized). With these, we numerically verify the landscape complexity of random ferromagnets, finding a diverging number of (1-spin-flip-stable) local minima as $N\to\infty$. We then investigate some of the properties of these local minima (e.g., typical energy and magnetization) and compare to the situation where the edge-weights are drawn from a heavy-tailed distribution.

cond-mat.dis-nn↗

arXiv2019

Nature vs. Nurture: Dynamical Evolution in Disordered Ising Ferromagnets

We study the predictability of zero-temperature Glauber dynamics in various models of disordered ferromagnets. This is analyzed using two independent dynamical realizations with the same random initialization (called twins). We derive, theoretically and numerically, trajectories for the evolution of the normalized magnetization and twin overlap as the system size tends to infinity. The systems we treat include mean-field ferromagnets with light-tailed and heavy-tailed coupling distributions, as well as highly-disordered models with a variety of other geometries. In the mean-field setting with light-tailed couplings, the disorder averages out and the limiting trajectories of the magnetization and twin overlap match those of the homogenous Curie--Weiss model. On the other hand, when the coupling distribution has heavy tails, or the geometry changes, the effect of the disorder persists in the thermodynamic limit. Nonetheless, qualitatively all such random ferromagnets share a similar time evolution for their twin overlap, wherein the two twins initially decorrelate, before either partially or fully converging back together due to the ferromagnetic drift.

cond-mat.stat-mech↗

arXiv2019

Exponential decay for the near-critical scaling limit of the planar Ising model

We consider the Ising model at its critical temperature with external magnetic field $ha^{15/8}$ on the square lattice with lattice spacing $a$. We show that the truncated two-point function in this model decays exponentially with a rate independent of $a$. As a consequence, we show exponential decay in the near-critical scaling limit Euclidean magnetization field. For the lattice model with $a=1$, the mass (inverse correlation length) is of order $h^{8/15}$ as $h\downarrow 0$; for the Euclidean field, it equals exactly $Ch^{8/15}$ for some $C$. Although there has been much progress in the study of critical scaling limits, results on near-critical models are far fewer due to the lack of conformal invariance away from the critical point. Our arguments combine lattice and continuum FK representations, including coupled conformal loop and measure ensembles, showing that such ensembles can be useful even in the study of near-critical scaling limits. Thus we provide the first substantial application of measure ensembles.

math.PR↗

arXiv2019

Constants of de Bruijn-Newman type in analytic number theory and statistical physics

One formulation in 1859 of the Riemann Hypothesis (RH) was that the Fourier transform $H_f(z)$ of $f$ for $ z \in \mathbb{C}$ has only real zeros when $f(t)$ is a specific function $Φ(t)$. Pólya's 1920s approach to RH extended $H_f$ to $H_{f,λ}$, the Fourier transform of $e^{λt^2} f(t)$. We review developments of this approach to RH and related ones in statistical physics where $f(t)$ is replaced by a measure $d ρ(t)$. Pólya's work together with 1950 and 1976 results of de Bruijn and Newman, respectively, imply the existence of a finite constant $Λ_{DN} = Λ_{DN} (Φ)$ in $(-\infty, 1/2]$ such that $H_{Φ,λ}$ has only real zeros if and only if $λ\geq Λ_{DN}$; RH is then equivalent to $Λ_{DN} \leq 0$. Recent developments include the Rodgers and Tao proof of the 1976 conjecture that $Λ_{DN} \geq 0$ (that RH, if true, is only barely so) and the Polymath 15 project improving the $1/2$ upper bound to about $0.22$. We also present examples of $ρ$'s with differing $H_{ρ,λ}$ and $Λ_{DN} (ρ)$ behaviors; some of these are new and based on a recent weak convergence theorem of the authors.

math.PR↗

arXiv2019

FK-Ising coupling applied to near-critical planar models

We consider the Ising model at its critical temperature with external magnetic field $ha^{15/8}$ on $a\mathbb{Z}^2$. We give a purely probabilistic proof, using FK methods rather than reflection positivity, that for $a=1$, the correlation length is $\geq const.~h^{-8/15}$ as $h\downarrow0$. We extend to the $a\downarrow0$ continuum limit the FK-Ising coupling for all $h>0$, and obtain tail estimates for the largest renormalized cluster area in a finite domain as well as an upper bound with exponent $1/8$ for the one-arm event. Finally, we show that for $a=1$, the average magnetization, $\mathcal{M}(h)$, in $\mathbb{Z}^2$ satisfies $\mathcal{M}(h)/h^{1/15}\rightarrow$ some $B\in(0,\infty)$ as $h\downarrow0$.

math.PR↗

arXiv2018

A note on exponential decay in the random field Ising model

For the two-dimensional random field Ising model (RFIM) with bimodal (i.e., two-valued) external field, we prove exponential decay of correlations either (1) when the temperature is larger than the critical temperature of the Ising model without external field and the magnetic field strength is small or (2) at any temperature when the magnetic field strength is sufficiently large. Unlike previous work on exponential decay, our approach is not based on cluster expansions but rather on arguably simpler methods; these combine an analysis of the Kertész line and a coupling of Ising measures (and also their random cluster representations) with different boundary conditions. We also show similar but weaker results for the RFIM with a general field distribution and in any dimension.

math.PR↗

arXiv2018

Lee-Yang Property and Gaussian multiplicative chaos

The Lee-Yang property of certain moment generating functions having only pure imaginary zeros is valid for Ising type models with one-component spins and XY models with two-component spins. Villain models and complex Gaussian multiplicative chaos are two-component systems analogous to XY models and related to Gaussian free fields. Although the Lee-Yang property is known to be valid generally in the first case, we show that is not so in the second. Our proof is based on two theorems of general interest relating the Lee-Yang property to distribution tail behavior.

math-ph↗

arXiv2017

A continuum of pure states in the Ising model on a halfplane

We study the homogeneous nearest-neighbor Ising ferromagnet on the right half plane with a Dobrushin type boundary condition --- say plus on the top part of the boundary and minus on the bottom. For sufficiently low temperature $T$, we completely characterize the pure (i.e., extremal) Gibbs states, as follows. There is exactly one for each angle $θ\in\lbrack-π/2,+π/2]$; here $θ$ specifies the asymptotic angle of the interface separating regions where the spin configuration looks like that of the plus (respectively, minus) full-plane state. Some of these conclusions are extended all the way to $T=T_{c}$ by developing new Ising exact solution result -- in particular, there is at least one pure state for each $θ$.

math-ph↗

arXiv2017

Fixation for Distributed Clustering Processes

We study a discrete-time resource flow in $Z^d$, where wealthier vertices attract the resources of their less rich neighbors. For any translation-invariant probability distribution of initial resource quantities, we prove that the flow at each vertex terminates after finitely many steps. This answers (a generalized version of) a question posed by van den Berg and Meester in 1991. The proof uses the mass-transport principle and extends to other graphs.

math.PR↗