Rigorous lower bound of the dynamical critical exponent of the Ising model

• URL: http://arxiv.org/abs/2502.09908v2
• Date: Wed, 21 May 2025 19:26:11 GMT
• Title: Rigorous lower bound of the dynamical critical exponent of the Ising model
• Authors: Rintaro Masaoka, Tomohiro Soejima, Haruki Watanabe,
• Abstract summary: We study the kinetic Ising model under Glauber dynamics.<n>We rigorously improve the previously known estimate $z geq 2 - eta$.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study the kinetic Ising model under Glauber dynamics and establish an upper bound on the spectral gap for finite systems. This bound implies the critical exponent inequality $z \geq 2$, thereby rigorously improving the previously known estimate $z \geq 2 - \eta$. Our proof relies on the mapping from stochastic processes to frustration-free quantum systems and leverages the Simon--Lieb and Gosset--Huang inequalities.

Related papers

• Lie symmetries and ghost-free representations of the Pais-Uhlenbeck model [44.99833362998488]
  We investigate the Pais-Uhlenbeck (PU) model, a paradigmatic example of a higher time-derivative theory.<n>Exploiting Lie symmetries in conjunction with the model's Bi-Hamiltonian structure, we construct distinct Poisson bracket formulations.<n>Our approach yields a unified framework for interpreting and stabilising higher time-derivative dynamics.
  arXiv  Detail & Related papers  (2025-05-09T15:16:40Z)
• Critical Dynamics in Short-Range Quadratic Hamiltonians [0.0]
  We consider critical dynamics that emerges when the Thouless time, i.e., the saturation time of the mean-squared displacement, approaches the typical Heisenberg time. We establish a relation, $z=d_l/d_s$, linking the critical dynamical exponent $z$ to $d_l$ and to the spectral fractal dimension $d_s$.
  arXiv  Detail & Related papers  (2025-03-04T17:51:40Z)
• Two-Point Deterministic Equivalence for Stochastic Gradient Dynamics in Linear Models [76.52307406752556]
  We derive a novel deterministic equivalence for the two-point function of a random resolvent.<n>We give a unified derivation of the performance of a wide variety of high-dimensional trained linear models with gradient descent.
  arXiv  Detail & Related papers  (2025-02-07T16:45:40Z)
• Criteria and Bias of Parameterized Linear Regression under Edge of Stability Regime [38.134523847923646]
  Edge of Stability (EoS) is usually known as the Edge of Stability (EoS) phenomenon.<n>We show that EoS occurs even when $l$ is quadratic under proper conditions.<n>We also shed some new light on the implicit bias of diagonal linear networks when a larger step-size is employed.
  arXiv  Detail & Related papers  (2024-12-11T02:07:37Z)
• Rigorous lower bound on dynamical exponents in gapless frustration-free systems [0.0]
  This work rigorously establishes a universal lower bound $zge2$ for the dynamical exponent in frustration-free quantum many-body systems.<n>Remarkably, our result can be applied to prove new bounds for dynamics of classical processes.
  arXiv  Detail & Related papers  (2024-06-10T16:08:33Z)
• Emergent criticality in a constrained boson model [0.0]
  We show, via explicit computation on a constrained bosonic model, that the presence of subsystem symmetries can lead to a quantum phase transition.<n>We verify this scenario via explicit exact-diagonalization computations, provide an effective Landau-Ginzburg theory for such a transition, and discuss the connection of our model to the PXP model describing Rydberg atom arrays.
  arXiv  Detail & Related papers  (2023-11-20T19:00:03Z)
• Entanglement dynamics in the many-body Hatano-Nelson model [0.0]
  The entanglement dynamics in a non-Hermitian quantum system is studied numerically and analyzed from the viewpoint of quasiparticle picture. As opposed to an assertion of previous studies, the entanglement dynamics in this non-Hermitian quantum system is very different from the one in its Hermitian counterpart.
  arXiv  Detail & Related papers  (2023-08-06T10:12:41Z)
• Nonequilibrium critical dynamics of the two-dimensional $\pm J$ Ising model [0.023787965910387825]
  The $pm J$ Ising model is a simple frustrated spin model, where the exchange couplings independently take the discrete value $-J$ with probability $p$ and $+J$ with probability $1-p$. Here, we investigate the nonequilibrium critical behavior of the two-dimensional $pm J$ Ising model, after a quench from different initial conditions to a critical point $T_c(p)$ on the paramagnetic-ferromagnetic (PF) transition line.
  arXiv  Detail & Related papers  (2023-04-24T11:02:33Z)
• Distorted stability pattern and chaotic features for quantized prey-predator-like dynamics [0.0]
  Non-equilibrium and instability features of prey-predator-like systems are investigated in the framework of the Weyl-Wigner quantum mechanics. From the non-Liouvillian pattern driven by the associated Wigner currents, hyperbolic equilibrium and stability parameters are shown to be affected by quantum distortions.
  arXiv  Detail & Related papers  (2023-03-16T19:55:36Z)
• Gradient-Free Methods for Deterministic and Stochastic Nonsmooth Nonconvex Optimization [94.19177623349947]
  Non-smooth non optimization problems emerge in machine learning and business making. Two core challenges impede the development of efficient methods with finitetime convergence guarantee. Two-phase versions of GFM and SGFM are also proposed and proven to achieve improved large-deviation results.
  arXiv  Detail & Related papers  (2022-09-12T06:53:24Z)
• Beyond the Edge of Stability via Two-step Gradient Updates [49.03389279816152]
  Gradient Descent (GD) is a powerful workhorse of modern machine learning. GD's ability to find local minimisers is only guaranteed for losses with Lipschitz gradients. This work focuses on simple, yet representative, learning problems via analysis of two-step gradient updates.
  arXiv  Detail & Related papers  (2022-06-08T21:32:50Z)
• KL-Entropy-Regularized RL with a Generative Model is Minimax Optimal [70.15267479220691]
  We consider and analyze the sample complexity of model reinforcement learning with a generative variance-free model. Our analysis shows that it is nearly minimax-optimal for finding an $varepsilon$-optimal policy when $varepsilon$ is sufficiently small.
  arXiv  Detail & Related papers  (2022-05-27T19:39:24Z)
• Global Convergence of Over-parameterized Deep Equilibrium Models [52.65330015267245]
  A deep equilibrium model (DEQ) is implicitly defined through an equilibrium point of an infinite-depth weight-tied model with an input-injection. Instead of infinite computations, it solves an equilibrium point directly with root-finding and computes gradients with implicit differentiation. We propose a novel probabilistic framework to overcome the technical difficulty in the non-asymptotic analysis of infinite-depth weight-tied models.
  arXiv  Detail & Related papers  (2022-05-27T08:00:13Z)
• Second Order Path Variationals in Non-Stationary Online Learning [23.91519151164528]
  We show that appropriately designed Strongly Adaptive algorithms achieve a dynamic regret of $tilde O(d2 n1/5 C_n2/5 vee d2)$. The aforementioned dynamic regret rate is shown to be optimal modulo dimension dependencies and poly-logarithmic factors of $n$.
  arXiv  Detail & Related papers  (2022-05-04T07:14:39Z)
• Improved Convergence Rate of Stochastic Gradient Langevin Dynamics with Variance Reduction and its Application to Optimization [50.83356836818667]
  gradient Langevin Dynamics is one of the most fundamental algorithms to solve non-eps optimization problems. In this paper, we show two variants of this kind, namely the Variance Reduced Langevin Dynamics and the Recursive Gradient Langevin Dynamics.
  arXiv  Detail & Related papers  (2022-03-30T11:39:00Z)
• Single Trajectory Nonparametric Learning of Nonlinear Dynamics [8.438421942654292]
  Given a single trajectory of a dynamical system, we analyze the performance of the nonparametric least squares estimator (LSE) We leverage recently developed information-theoretic methods to establish the optimality of the LSE for non hypotheses classes. We specialize our results to a number of scenarios of practical interest, such as Lipschitz dynamics, generalized linear models, and dynamics described by functions in certain classes of Reproducing Kernel Hilbert Spaces (RKHS)
  arXiv  Detail & Related papers  (2022-02-16T19:38:54Z)
• Decimation technique for open quantum systems: a case study with driven-dissipative bosonic chains [62.997667081978825]
  Unavoidable coupling of quantum systems to external degrees of freedom leads to dissipative (non-unitary) dynamics. We introduce a method to deal with these systems based on the calculation of (dissipative) lattice Green's function. We illustrate the power of this method with several examples of driven-dissipative bosonic chains of increasing complexity.
  arXiv  Detail & Related papers  (2022-02-15T19:00:09Z)
• Convex Analysis of the Mean Field Langevin Dynamics [49.66486092259375]
  convergence rate analysis of the mean field Langevin dynamics is presented. $p_q$ associated with the dynamics allows us to develop a convergence theory parallel to classical results in convex optimization.
  arXiv  Detail & Related papers  (2022-01-25T17:13:56Z)
• Policy Optimization Using Semiparametric Models for Dynamic Pricing [1.3428344011390776]
  We study the contextual dynamic pricing problem where the market value of a product is linear in its observed features plus some market noise. We propose a dynamic statistical learning and decision-making policy that combines semiparametric estimation from a generalized linear model with an unknown link and online decision-making.
  arXiv  Detail & Related papers  (2021-09-13T23:50:01Z)
• Hyperbolic Variational Graph Neural Network for Modeling Dynamic Graphs [77.33781731432163]
  We learn dynamic graph representation in hyperbolic space, for the first time, which aims to infer node representations. We present a novel Hyperbolic Variational Graph Network, referred to as HVGNN. In particular, to model the dynamics, we introduce a Temporal GNN (TGNN) based on a theoretically grounded time encoding approach.
  arXiv  Detail & Related papers  (2021-04-06T01:44:15Z)
• Continuous-time quantum walks in the presence of a quadratic perturbation [55.41644538483948]
  We address the properties of continuous-time quantum walks with Hamiltonians of the form $mathcalH= L + lambda L2$. We consider cycle, complete, and star graphs because paradigmatic models with low/high connectivity and/or symmetry.
  arXiv  Detail & Related papers  (2020-05-13T14:53:36Z)
• Robust $k$-means Clustering for Distributions with Two Moments [4.21934751979057]
  We consider the robust algorithms for the $k$-means clustering problem where a quantizer is constructed based on $N$ independent observations. Our main results are median of means based non-asymptotic excess distortion bounds that hold under the two bounded moments assumption in a general separable Hilbert space.
  arXiv  Detail & Related papers  (2020-02-06T16:36:53Z)
• On the Difference Between the Information Bottleneck and the Deep Information Bottleneck [81.89141311906552]
  We revisit the Deep Variational Information Bottleneck and the assumptions needed for its derivation. We show how to circumvent this limitation by optimising a lower bound for $I(T;Y)$ for which only the latter Markov chain has to be satisfied.
  arXiv  Detail & Related papers  (2019-12-31T18:31:42Z)

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.