Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA Relation between Disorder Chaos and Incongruent States in Spin Glasses on ${mathbb Z}^d$
58
0
0.0
(
0
)
Added by
Louis-Pierre Arguin
Publication date
2018
fields
Physics
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
We present a detailed proof of a previously announced result (C.M. Newman and D.L. Stein, Phys. Rev. Lett. v. 84, pp. 3966--3969 (2000)) supporting the absence of multiple (incongruent) ground state pairs for 2D Edwards-Anderson spin glasses (with zero external field and, e.g., Gaussian couplings): if two ground state pairs (chosen from metastates with, e.g., periodic boundary conditions) on the infinite square lattice are distinct, then the dual bonds where they differ form a single doubly-infinite, positive-density domain wall. It is an open problem to prove that such a situation cannot occur (or else to show --- much less likely in our opinion --- that it indeed does happen) in these models. Our proof involves an analysis of how (infinite-volume) ground states change as (finitely many) couplings vary, which leads us to a notion of zero-temperature excitation metastates, that may be of independent interest.
A two-type version of the frog model on $mathbb{Z}^d$ is formulated, where active type $i$ particles move according to lazy random walks with probability $p_i$ of jumping in each time step ($i=1,2$). Each site is independently assigned a random number of particles. At time 0, the particles at the origin are activated and assigned type 1 and the particles at one other site are activated and assigned type 2, while all other particles are sleeping. When an active type $i$ particle moves to a new site, any sleeping particles there are activated and assigned type $i$, with an arbitrary tie-breaker deciding the type if the site is hit by particles of both types in the same time step. We show that the event $G_i$ that type $i$ activates infinitely many particles has positive probability for all $p_1,p_2in(0,1]$ ($i=1,2$). Furthermore, if $p_1=p_2$, then the types can coexist in the sense that $mathbb{P}(G_1cap G_2)>0$. We also formulate several open problems. For instance, we conjecture that, when the initial number of particles per site has a heavy tail, the types can coexist also when $p_1 eq p_2$.
We discuss temperature chaos in mean field and realistic 3D spin glasses. Our numerical simulations show no trace of a temperature chaotic behavior for the system sizes considered. We discuss the experimental and theoretical implications of these findings.
Some stochastic systems are particularly interesting as they exhibit critical behavior without fine-tuning of a parameter, a phenomenon called self-organized criticality. In the context of driven-dissipative steady states, one of the main models is that of Activated Random Walks. Long-range effects intrinsic to the conservative dynamics and lack of a simple algebraic structure cause standard tools and techniques to break down. This makes the mathematical study of this model remarkably challenging. Yet, some exciting progress has been made in the last ten years, with the development of a framework of tools and methods which is finally becoming more structured. In these lecture notes we present the existing results and reproduce the techniques developed so far.
Mean-field spin glasses are families of random energy functions (Hamiltonians) on high-dimensional product spaces. In this paper we consider the case of Ising mixed $p$-spin models, namely Hamiltonians $H_N:Sigma_Nto {mathbb R}$ on the Hamming hypercube $Sigma_N = {pm 1}^N$, which are defined by the property that ${H_N({boldsymbol sigma})}_{{boldsymbol sigma}in Sigma_N}$ is a centered Gaussian process with covariance ${mathbb E}{H_N({boldsymbol sigma}_1)H_N({boldsymbol sigma}_2)}$ depending only on the scalar product $langle {boldsymbol sigma}_1,{boldsymbol sigma}_2rangle$. The asymptotic value of the optimum $max_{{boldsymbol sigma}in Sigma_N}H_N({boldsymbol sigma})$ was characterized in terms of a variational principle known as the Parisi formula, first proved by Talagrand and, in a more general setting, by Panchenko. The structure of superlevel sets is extremely rich and has been studied by a number of authors. Here we ask whether a near optimal configuration ${boldsymbol sigma}$ can be computed in polynomial time. We develop a message passing algorithm whose complexity per-iteration is of the same order as the complexity of evaluating the gradient of $H_N$, and characterize the typical energy value it achieves. When the $p$-spin model $H_N$ satisfies a certain no-overlap gap assumption, for any $varepsilon>0$, the algorithm outputs ${boldsymbol sigma}inSigma_N$ such that $H_N({boldsymbol sigma})ge (1-varepsilon)max_{{boldsymbol sigma}} H_N({boldsymbol sigma})$, with high probability. The number of iterations is bounded in $N$ and depends uniquely on $varepsilon$. More generally, regardless of whether the no-overlap gap assumption holds, the energy achieved is given by an extended variational principle, which generalizes the Parisi formula.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
