Dynamical phases in a "multifractal" Rosenzweig-Porter model
- URL: http://arxiv.org/abs/2106.01965v2
- Date: Fri, 11 Jun 2021 18:25:28 GMT
- Title: Dynamical phases in a "multifractal" Rosenzweig-Porter model
- Authors: I. M. Khaymovich, V. E. Kravtsov
- Abstract summary: We present a general theory of survival probability in a random-matrix model.
We identify the exponential, the stretch-exponential and the frozen-dynamics phases.
Our theory allows to compute the shift of apparent phase transition lines at a finite system size.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: We consider the static and dynamic phases in a Rosenzweig-Porter (RP) random
matrix ensemble with the tailed distribution of off-diagonal matrix elements of
the form of the large-deviation ansatz. We present a general theory of survival
probability in such a random-matrix model and show that the {\it averaged}
survival probability may decay with time as the simple exponent, as the
stretch-exponent and as a power-law or slower. Correspondingly, we identify the
exponential, the stretch-exponential and the frozen-dynamics phases. As an
example, we consider the mapping of the Anderson model on Random Regular Graph
(RRG) onto the "multifractal" RP model and find exact values of the
stretch-exponent $\kappa$ depending on box-distributed disorder in the
thermodynamic limit. As another example we consider the logarithmically-normal
RP (LN-RP) random matrix ensemble and find analytically its phase diagram and
the exponent $\kappa$. In addition, our theory allows to compute the shift of
apparent phase transition lines at a finite system size and show that in the
case of RP associated with RRG and LN-RP with the same symmetry of distribution
function of hopping, a finite-size multifractal "phase" emerges near the
tricritical point which is also the point of localization transition.
Related papers
- Single-Particle Universality of the Many-Body Spectral Form Factor [0.0]
We consider systems of fermions evolved by non-interacting unitary circuits with correlated on-site potentials.
When these potentials are drawn from the eigenvalue distribution of a circular random matrix ensemble, the spectral form factor (SFF) of the resulting circuit ensemble can be computed exactly without numerical sampling.
arXiv Detail & Related papers (2024-10-09T18:00:00Z) - KPZ scaling from the Krylov space [83.88591755871734]
Recently, a superdiffusion exhibiting the Kardar-Parisi-Zhang scaling in late-time correlators and autocorrelators has been reported.
Inspired by these results, we explore the KPZ scaling in correlation functions using their realization in the Krylov operator basis.
arXiv Detail & Related papers (2024-06-04T20:57:59Z) - Theory of mobility edge and non-ergodic extended phase in coupled random
matrices [18.60614534900842]
The mobility edge, as a central concept in disordered models for localization-delocalization transitions, has rarely been discussed in the context of random matrix theory.
We show that their overlapped spectra and un-overlapped spectra exhibit totally different scaling behaviors, which can be used to construct tunable mobility edges.
Our model provides a general framework to realize the mobility edges and non-ergodic phases in a controllable way in RMT.
arXiv Detail & Related papers (2023-11-15T01:43:37Z) - Anatomy of the eigenstates distribution: a quest for a genuine
multifractality [0.0]
Interest in multifractal phases has risen as they are believed to be present in the Many-Body Localized (MBL) phase.
Several RP-like ensembles with the fat-tailed distributed hopping terms have been proposed, with claims that they host the desired multifractal phase.
arXiv Detail & Related papers (2023-09-12T18:00:01Z) - Adaptive Annealed Importance Sampling with Constant Rate Progress [68.8204255655161]
Annealed Importance Sampling (AIS) synthesizes weighted samples from an intractable distribution.
We propose the Constant Rate AIS algorithm and its efficient implementation for $alpha$-divergences.
arXiv Detail & Related papers (2023-06-27T08:15:28Z) - Machine learning in and out of equilibrium [58.88325379746631]
Our study uses a Fokker-Planck approach, adapted from statistical physics, to explore these parallels.
We focus in particular on the stationary state of the system in the long-time limit, which in conventional SGD is out of equilibrium.
We propose a new variation of Langevin dynamics (SGLD) that harnesses without replacement minibatching.
arXiv Detail & Related papers (2023-06-06T09:12:49Z) - A Probabilistic Interpretation of Transformers [91.3755431537592]
We propose a probabilistic interpretation of exponential dot product attention of transformers and contrastive learning based off of exponential families.
We state theoretical limitations of our theory and the Hopfield theory and suggest directions for resolution.
arXiv Detail & Related papers (2022-04-28T23:05:02Z) - Complexity in the Lipkin-Meshkov-Glick Model [0.0]
We study complexity in a spin system with infinite range interaction.
Exact expressions for the Nielsen complexity (NC) and the Fubini-Study complexity (FSC) are derived.
arXiv Detail & Related papers (2022-04-13T13:11:58Z) - Emergent fractal phase in energy stratified random models [0.0]
We study the effects of partial correlations in kinetic hopping terms of long-range random matrix models on their localization properties.
We show that any deviation from the completely correlated case leads to the emergent non-ergodic delocalization in the system.
arXiv Detail & Related papers (2021-06-07T18:00:01Z) - Graph Gamma Process Generalized Linear Dynamical Systems [60.467040479276704]
We introduce graph gamma process (GGP) linear dynamical systems to model real multivariate time series.
For temporal pattern discovery, the latent representation under the model is used to decompose the time series into a parsimonious set of multivariate sub-sequences.
We use the generated random graph, whose number of nonzero-degree nodes is finite, to define both the sparsity pattern and dimension of the latent state transition matrix.
arXiv Detail & Related papers (2020-07-25T04:16:34Z) - Kernel and Rich Regimes in Overparametrized Models [69.40899443842443]
We show that gradient descent on overparametrized multilayer networks can induce rich implicit biases that are not RKHS norms.
We also demonstrate this transition empirically for more complex matrix factorization models and multilayer non-linear networks.
arXiv Detail & Related papers (2020-02-20T15:43:02Z)
This list is automatically generated from the titles and abstracts of the papers in this site.
This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.