We study the zero-temperature Glauber dynamics of homogeneous Ising ferromagnets on hypercubes, as their dimension d varies. By hypercube, we mean the d-dimensional hypercubic graph with 2^{d} vertices, where each vertex is connected to d neighbors. We investigate the asymptotic (d→∞ and time t→∞) 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). Blinkers 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 versus 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.

We complete the verification of the 1952 Yang and Lee proposal that thermodynamic singularities are exactly the limits in R${\mathbb {R}}$ of finite-volume singularities in C${\mathbb {C}}$. For the Ising model defined on a finite Lambda subset of Zd$\Lambda \subset \mathbb {Z}<^>d$ at inverse temperature beta >= 0$\beta \ge 0$ and external field h, let alpha 1(Lambda,beta)$\alpha _1(\Lambda ,\beta )$ be the modulus of the first zero (that closest to the origin) of its partition function (in the variable h). We prove that alpha 1(Lambda,beta)$\alpha _1(\Lambda ,\beta )$ decreases to alpha 1(Zd,beta)$\alpha _1(\mathbb {Z}<^>d,\beta )$ as Lambda increases to Zd$\mathbb {Z}<^>d$ where alpha 1(Zd,beta)is an element of[0,infinity)$\alpha _1(\mathbb {Z}<^>d,\beta )\in [0,\infty )$ 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 alpha 1(Zd,beta)$\alpha _1(\mathbb {Z}<^>d,\beta )$ is strictly positive if and only if beta is strictly less than the critical inverse temperature.

We investigate scenarios in which the low-temperature phase of short-range spin glasses comprises thermodynamic states which are nontrivial mixtures of multiple incongruent pure state pairs. We construct a new kind of metastate supported on Gibbs states whose edge overlap values with a reference state fall within a specified range. Using this metastate we show that, in any dimension, the variance of free energy difference fluctuations between pure states within a single mixed Gibbs state with multiple edge overlap values diverges linearly with the volume. We discuss some implications of these results.

We show that the notion of critical droplets is central to an understanding of the nature of ground states in the Edwards-Anderson Ising model of a spin glass in arbitrary dimension. Given a specific ground state, suppose the coupling value for a given edge is varied with all other couplings held fixed. Beyond some specific value of the coupling, a droplet will flip leading to a new ground state; we refer to this as the critical droplet for that edge and ground state. We show that the distribution of sizes and energies over all edges for a specific ground state can be used to determine which of the leading scenarios for the spin glass phase is correct. In particular, the existence of low-energy interfaces between incongruent ground states as predicted by replica symmetry breaking is equivalent to the presence of critical droplets whose boundaries comprise a positive fraction of edges in the infinite lattice.

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 $$t\ge 0$$ (i.e., the Curie-Weiss model), we prove that all the principal zeros (those in $$i[0,\pi /2)$$ ) decrease strictly in t.

In this review we define and discuss metastates, mathematical tools with general applicability to thermodynamic systems which are particularly useful when working with disordered or inhomogeneous short-range systems. In an infinite such system there may be many competing thermodynamic states, which can lead to the absence of a straightforward thermodynamic limit of local correlation functions. A metastate is a probability measure on the infinite-volume thermodynamic states that restores the connection between those states and the Gibbs states observed in finite volumes. After introducing the basic metastates and discussing their properties, we present possible scenarios for the spin-glass phase and discuss what the metastate approach reveals about how replica symmetry breaking would manifest itself in finite-dimensional short-range spin glasses.

We consider short-range Ising spin glasses in equilibrium at infinite system size, and prove that, for fixed bond realization and a given Gibbs state drawn from a suitable metastate, each translation- and locally-invariant function (for example, self-overlaps) of a single pure state in the decomposition of the Gibbs state takes the same value for all the pure states in that Gibbs state. We describe several significant applications to spin glasses.

In this paper, we consider Ising models with ferromagnetic pair interactions. We prove that the Ursell functions u_2k satisfy: (-1)^k-1u_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.

We propose an approach toward understanding the spin glass phase at zero and low temperature by studying the stability of a spin glass ground state against perturbations of a single coupling. After reviewing the concepts of flexibility, critical droplet, and related quantities for both finite- and infinite-volume ground states, we study some of their properties and review three models in which these quantities are partially or fully understood. We also review a recent result showing the connection between our approach and that of disorder chaos. We then view four proposed scenarios for the low-temperature spin glass phase-replica symmetry breaking, scaling-droplet, TNT, and chaotic pairs-through the lens of the predictions of each scenario for the lowest-energy large-lengthscale excitations above the ground state. Using a new concept called σ-criticality, which quantifies the sensitivity of ground states to single-bond coupling variations, we show that each of these four pictures can be identified with different critical droplet geometries and energies. We also investigate necessary and sufficient conditions for the existence of multiple incongruent ground states.

Consider the nearest-neighbor Ising model on $$\Lambda _n:=[-n,n]^d\cap {\mathbb {Z}}^d$$ at inverse temperature $$\beta \ge 0$$ with free boundary conditions, and let $$Y_n(\sigma ):=\sum _{u\in \Lambda _n}\sigma _u$$ be its total magnetization. Let $$X_n$$ be the total magnetization perturbed by a critical Curie–Weiss interaction, i.e., $$\begin{aligned} \frac{d F_{X_n}}{d F_{Y_n}}(x):=\frac{\exp [x^2/\left( 2\langle Y_n^2 \rangle _{\Lambda _n,\beta }\right) ]}{\left\langle \exp [Y_n^2/\left( 2\langle Y_n^2\rangle _{\Lambda _n,\beta }\right) ]\right\rangle _{\Lambda _n,\beta }}, \end{aligned}$$ where $$F_{X_n}$$ and $$F_{Y_n}$$ are the distribution functions for $$X_n$$ and $$Y_n$$ respectively. We prove that for any $$d\ge 4$$ and $$\beta \in [0,\beta _c(d)]$$ where $$\beta _c(d)$$ 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 $$d\ge 1$$ and then for $$\beta $$ small, $$\begin{aligned} f_X(x)=K\exp (-C^4x^4), \end{aligned}$$ where $$C=\sqrt{\Gamma (3/4)/\Gamma (1/4)}$$ and $$K=\sqrt{\Gamma (3/4)}/(4\Gamma (5/4)^{3/2})$$ . Possible connections between $$f_X$$ and the high-dimensional critical Ising model with periodic boundary conditions are discussed.

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.

We study the critical Ising model with free boundary conditions on finite domains in ℤ^d with d≥ 4 . 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→∞ . 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.

We review the construction of a stationary Gaussian process $X(t)$ starting from the near-critical continuum scaling limit $\Phi^h$ of the Ising magnetization and its relation to the mass spectrum of the relativistic quantum field theory associated to $\Phi^h$. 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 $d>2$.

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 $\Phi (t)$. Pólya's 1920s approach to RH extended $H_f$ to $H_{f,\lambda}$, the Fourier transform of $e^{\lambda t^2} f(t)$. We review developments of this approach to RH and related ones in statistical physics where $f(t)$ is replaced by a measure $d \rho (t)$. Pólya's work together with 1950 and 1976 results of de Bruijn and Newman, respectively, imply the existence of a finite constant $\Lambda_{DN} = \Lambda_{DN} (\Phi)$ in $(-\infty, 1/2]$ such that $H_{\Phi,\lambda}$ has only real zeros if and only if $\lambda \geq \Lambda_{DN}$; RH is then equivalent to $\Lambda_{DN} \leq 0$. Recent developments include the Rodgers and Tao proof of the 1976 conjecture that $\Lambda_{DN} \geq 0$ (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 $\rho$'s with differing $H_{\rho,\lambda}$ and $\Lambda_{DN} (\rho)$ behaviors; some of these are new and based on a recent weak convergence theorem of the authors.

We consider the Ising model at its critical temperature with external magnetic field ha15∕8 on aZ2. We give a purely probabilistic proof, using FK methods rather than reflection positivity, that for a=1, the correlation length is ≥const.h−8∕15 as h↓0. We extend to the a↓0 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, M(h), in Z2 satisfies M(h)∕h1∕15→ some B∈(0,∞) as h↓0.

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.

In independent bond percolation on $${\mathbb {Z}}^d$$ with parameter p, if one removes the vertices of the infinite cluster (and incident edges), for which values of p does the remaining graph contain an infinite connected component? Grimmett-Holroyd-Kozma used the triangle condition to show that for $$d \ge 19$$, the set of such p contains values strictly larger than the percolation threshold $$p_c$$. With the work of Fitzner-van der Hofstad, this has been reduced to $$d \ge 11$$. We improve this result by showing that for $$d \ge 10$$ and some $$p>p_c$$, there are infinite paths consisting of “shielded” vertices—vertices all whose adjacent edges are closed—which must be in the complement of the infinite cluster. Using values of $$p_c$$ obtained from computer simulations, this bound can be reduced to $$d \ge 7$$. Our methods are elementary and do not require the triangle condition.

An important but little-studied property of spin glasses is the stability of their ground states to changes in one or a finite number of couplings. It was shown in earlier work that, if multiple ground states are assumed to exist, then fluctuations in their energy differences --- and therefore the possibility of multiple ground states --- are closely related to the stability of their ground states. Here we examine the stability of ground states in two models, one of which is presumed to have a ground state structure that is qualitatively similar to other realistic short-range spin glasses in finite dimensions.

Let $$\Phi ^h(x)$$ with $$x=(t,y)$$ denote the near-critical scaling limit of the planar Ising magnetization field. We take the limit of $$\Phi ^h$$ 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 K(t) is the Laplace transform of a mass spectral measure $$\rho $$ of the relativistic quantum field theory associated to the Euclidean field $$\Phi ^h.$$X and K should provide a useful tool for studying the mass spectrum; e.g., the small distance/time behavior of the covariance functions of $$\Phi ^h$$ and X(t) shows that $$\rho $$ is finite but has infinite first moment.

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.