arXiv subjects

C. M. Newman

Publications and source records attributed to C. M. Newman.

At least 19 recordsLinked to original sources

arXiv2026

Ground State Excitations and Energy Fluctuations in Short-Range Spin Glasses

We study the stability of ground states in the Edwards-Anderson Ising spin glass in dimensions two and higher against perturbations of a single coupling. After reviewing the concepts of critical droplets, flexibilities and metastates, we show that, in any dimension, a certain kind of critical droplet with space-filling (i.e., positive spatial density) boundary does not exist in ground states generated by coupling-independent boundary conditions. Using this we show that if incongruent ground states exist in any dimension, the variance of their energy difference restricted to finite volumes scales proportionally to the volume. This in turn is used to prove that a metastate generated by (e.g.) periodic boundary conditions is unique and supported on a single pair of spin-reversed ground states in two dimensions. We further show that any excitation above an infinite-volume ground state with a positive-density interface must have the property that its energy difference with the ground state in a restricted volume diverges as the square root of the restricted volume.

cond-mat.dis-nn↗

arXiv2024

Critical Droplets and Replica Symmetry Breaking

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.

cond-mat.dis-nn↗

arXiv2024

Free Energy Difference Fluctuations in Short-Range Spin Glasses

It is generally believed (but not yet proved) that Ising spin glasses with nearest-neighbor interactions have a phase transition in three and higher dimensions to a low-temperature spin glass phase, but the nature of this phase remains controversial, especially whether it is characterized by multiple incongruent Gibbs states. Of particular relevance to this question is the behavior of the typical free energy difference restricted to a finite volume between two such putative Gibbs states, as well as the nature of the fluctuations of their free energy difference as the couplings within the volume vary. In this paper we investigate these free energy difference fluctuations by introducing a new kind of metastate which classifies Gibbs states through their edge overlap values with a reference Gibbs state randomly chosen from the support of the periodic boundary condition (PBC) metastate. We find that the free energy difference between any two incongruent pure states, regardless of the details of how they're organized into mixed states within the PBC metastate, converges to a Gaussian (or Gaussian-like) distribution whose variance scales with the volume, proving a decades-old conjecture of Fisher and Huse. The same conclusion applies, though with some additional restrictions, to both mixed Gibbs states and ground states. We discuss some implications of these results.

cond-mat.dis-nn↗

arXiv2024

Overlap Structure and Free Energy Fluctuations in Short-Range Spin Glasses

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.

cond-mat.dis-nn↗

arXiv2023

Proof of single-replica equivalence in 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.

cond-mat.dis-nn↗

arXiv2022

Metastates and Replica Symmetry Breaking

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.

cond-mat.dis-nn↗

arXiv2021

Ground State Stability and the Nature of the Spin Glass Phase

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 sigma-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.

cond-mat.dis-nn↗

arXiv2020

Ground state stability in two spin glass models

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.

cond-mat.dis-nn↗

arXiv2019

Percolation of finite clusters and shielded paths

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 cluster? Grimmett-Holroyd-Kozma used the triangle condition to show that for $d \geq 19$, 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 $d \geq 11$. We improve this result by showing that for $d \geq 10$ 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 numerical values of $p_c$, this bound can be reduced to $d \geq 7$. Our methods are elementary and do not require the triangle condition.

math.PR↗

arXiv2018

A Relation between Disorder Chaos and Incongruent States in Spin Glasses on ${\mathbb Z}^d$

We derive lower bounds for the variance of the difference of energies between incongruent ground states, i.e., states with edge overlaps strictly less than one, of the Edwards-Anderson model on ${\mathbb Z}^d$. The bounds highlight a relation between the existence of incongruent ground states and the absence of edge disorder chaos. In particular, it suggests that the presence of disorder chaos is necessary for the variance to be of order less than the volume. In addition, a relation is established between the scale of disorder chaos and the size of critical droplets. The results imply a long-conjectured relation between the droplet theory of Fisher and Huse and the absence of incongruence.

math.PR↗

arXiv2017

Long-Time Predictability in Disordered Spin Systems Following a Deep Quench

We study the problem of predictability, or "nature vs. nurture", in several disordered Ising spin systems evolving at zero temperature from a random initial state: how much does the final state depend on the information contained in the initial state, and how much depends on the detailed history of the system? Our numerical studies of the "dynamical order parameter" in Edwards-Anderson Ising spin glasses and random ferromagnets indicate that the influence of the initial state decays as dimension increases. Similarly, this same order parameter for the Sherrington-Kirkpatrick infinite-range spin glass indicates that this information decays as the number of spins increases. Based on these results, we conjecture that the influence of the initial state on the final state decays to zero in finite-dimensional random-bond spin systems as dimension goes to infinity, regardless of the presence of frustration. We also study the rate at which spins "freeze out" to a final state as a function of dimensionality and number of spins; here the results indicate that the number of "active" spins at long times increases with dimension (for short-range systems) or number of spins (for infinite-range systems). We provide theoretical arguments to support these conjectures, and also study analytically several mean-field models: the random energy model, the uniform Curie-Weiss ferromagnet, and the disordered Curie-Weiss ferromagnet. We find that for these models, the information contained in the initial state does not decay in the thermodynamic limit-- in fact, it fully determines the final state. Unlike in short-range models, the presence of frustration in mean-field models dramatically alters the dynamical behavior with respect to the issue of predictability.

cond-mat.stat-mech↗

arXiv2016

Perturbations of the Voter Model in One-Dimension

We study the scaling limit of a large class of voter model perturbations in one dimension, including stochastic Potts models, to a universal limiting object, the continuum voter model perturbation. The perturbations can be described in terms of bulk and boundary nucleations of new colors (opinions). The discrete and continuum (space) models are obtained from their respective duals, the discrete net with killing and Brownian net with killing. These determine the color genealogy by means of reduced graphs. We focus our attention on models where the voter and boundary nucleation dynamics depend only on the colors of nearest neighbor sites, for which convergence of the discrete net with killing to its continuum analog was proved in an earlier paper by the authors. We use some detailed properties of the Brownian net with killing to prove voter model perturbations convergence to its continuum counterpart. A crucial property of reduced graphs is that even in the continuum, they are finite almost surely. An important issue is how vertices of the continuum reduced graphs are strongly approximated by their discrete analogues.

math.PR↗

arXiv2016

Zero-Temperature Fluctuations in Short-Range Spin Glasses

We consider the energy difference restricted to a finite volume for certain pairs of incongruent ground states (if they exist) in the d-dimensional Edwards-Anderson (EA) Ising spin glass at zero temperature. We prove that the variance of this quantity with respect to the couplings grows at least proportionally to the volume in any dimension greater than or equal to two. An essential aspect of our result is the use of the excitation metastate. As an illustration of potential applications, we use this result to restrict the possible structure of spin glass ground states in two dimensions.

math-ph↗

arXiv2015

Coarsening with a frozen vertex

In the standard nearest-neighbor coarsening model with state space $\{-1,+1\}^{\mathbb{Z}^2}$ and initial state chosen from symmetric product measure, it is known (see~\cite{NNS}) that almost surely, every vertex flips infinitely often. In this paper, we study the modified model in which a single vertex is frozen to $+1$ for all time, and show that every other site still flips infinitely often. The proof combines stochastic domination (attractivity) and influence propagation arguments.

math.PR↗

arXiv2015

Universal Protein Distributions in a Model of Cell Growth and Division

Protein distributions measured under a broad set of conditions in bacteria and yeast were shown to exhibit a common skewed shape, with variances depending quadratically on means. For bacteria these properties were reproduced by temporal measurements of protein content, showing accumulation and division across generations. Here we present a stochastic growth-and-division model with feedback which captures these observed properties. The limiting copy number distribution is calculated exactly, and a single parameter is found to determine the distribution shape and the variance-to-mean relation. Estimating this parameter from bacterial temporal data reproduces the measured distribution shape with high accuracy, and leads to predictions for future experiments.

physics.bio-ph↗

arXiv2015

Thermodynamic Identities and Symmetry Breaking in Short-Range Spin Glasses

We present a technique to generate relations connecting pure state weights, overlaps, and correlation functions in short-range spin glasses. These are obtained directly from the unperturbed Hamiltonian and hold for general coupling distributions. All are satisfied in phases with simple thermodynamic structure, such as the droplet-scaling and chaotic pairs pictures. If instead nontrivial mixed-state pictures hold, the relations suggest that replica symmetry is broken as described by a Derrida-Ruelle cascade, with pure state weights distributed as a Poisson-Dirichlet process.

cond-mat.dis-nn↗

arXiv2014

Nature versus Nurture in Complex and Not-So-Complex Systems

Understanding the dynamical behavior of many-particle systems both in and out of equilibrium is a central issue in both statistical mechanics and complex systems theory. One question involves "nature versus nurture": given a system with a random initial state evolving through a well-defined stochastic dynamics, how much of the information contained in the state at future times depends on the initial condition ("nature") and how much on the dynamical realization ("nurture")? We discuss this question and present both old and new results for low-dimensional Ising spin systems.

cond-mat.stat-mech↗

arXiv2013

Nature vs. Nurture: Predictability in Low-Temperature Ising Dynamics

Consider a dynamical many-body system with a random initial state subsequently evolving through stochastic dynamics. What is the relative importance of the initial state ("nature") vs. the realization of the stochastic dynamics ("nurture") in predicting the final state? We examined this question for the two-dimensional Ising ferromagnet following an initial deep quench from $T=\infty$ to $T=0$. We performed Monte Carlo studies on the overlap between "identical twins" raised in independent dynamical environments, up to size $L=500$. Our results suggest an overlap decaying with time as $t^{-θ_h}$ with $θ_h = 0.22 \pm 0.02$; the same exponent holds for a quench to low but nonzero temperature. This "heritability exponent" may equal the persistence exponent for the 2D Ising ferromagnet, but the two differ more generally.

cond-mat.stat-mech↗