The Experts below are selected from a list of 288 Experts worldwide ranked by ideXlab platform
Slava Rychkov - One of the best experts on this subject based on the ideXlab platform.
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conformal invariance in the long range Ising Model
Nuclear Physics, 2016Co-Authors: Miguel F Paulos, Slava Rychkov, Balt C Van ReesAbstract:We consider the question of conformal invariance of the long-range Ising Model at the critical point. The continuum description is given in terms of a nonlocal field theory, and the absence of a stress tensor invalidates all of the standard arguments for the enhancement of scale invariance to conformal invariance. We however show that several correlation functions, computed to second order in the epsilon expansion, are nontrivially consistent with conformal invariance. We proceed to give a proof of conformal invariance to all orders in the epsilon expansion, based on the description of the long-range Ising Model as a defect theory in an auxiliary higher-dimensional space. A detailed review of conformal invariance in the d-dimensional short-range Ising Model is also included and may be of independent interest.
Balt C Van Rees - One of the best experts on this subject based on the ideXlab platform.
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conformal invariance in the long range Ising Model
Nuclear Physics, 2016Co-Authors: Miguel F Paulos, Slava Rychkov, Balt C Van ReesAbstract:We consider the question of conformal invariance of the long-range Ising Model at the critical point. The continuum description is given in terms of a nonlocal field theory, and the absence of a stress tensor invalidates all of the standard arguments for the enhancement of scale invariance to conformal invariance. We however show that several correlation functions, computed to second order in the epsilon expansion, are nontrivially consistent with conformal invariance. We proceed to give a proof of conformal invariance to all orders in the epsilon expansion, based on the description of the long-range Ising Model as a defect theory in an auxiliary higher-dimensional space. A detailed review of conformal invariance in the d-dimensional short-range Ising Model is also included and may be of independent interest.
Hidetoshi Nishimori - One of the best experts on this subject based on the ideXlab platform.
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quantum annealing in the transverse Ising Model
Physical Review E, 1998Co-Authors: Tadashi Kadowaki, Hidetoshi NishimoriAbstract:We introduce quantum fluctuations into the simulated annealing process of optimization problems, aiming at faster convergence to the optimal state. Quantum fluctuations cause transitions between states and thus play the same role as thermal fluctuations in the conventional approach. The idea is tested by the transverse Ising Model, in which the transverse field is a function of time similar to the temperature in the conventional method. The goal is to find the ground state of the diagonal part of the Hamiltonian with high accuracy as quickly as possible. We have solved the time-dependent Schrodinger equation numerically for small size systems with various exchange interactions. Comparison with the results of the corresponding classical ~thermal! method reveals that the quantum annealing leads to the ground state with much larger probability in almost all cases if we use the same annealing schedule. @S1063-651X~98!02910-9# specific Model system, rather than to develop a general argu- ment, to gain insight into the role of quantum fluctuations in the situation of optimization problem. Quantum effects have been found to play a very similar role to thermal fluctuations in the Hopfield Model in a transverse field in thermal equi- librium @5#. This observation motivates us to investigate dy- namical properties of the Ising Model under quantum fluc- tuations in the form of a transverse field. We therefore discuss in this paper the transverse Ising Model with a vari- ety of exchange interactions. The transverse field controls the rate of transition between states and thus plays the same role as the temperature does in SA. We assume that the system has no thermal fluctuations in the QA context and the term ''ground state'' refers to the lowest-energy state of the Hamiltonian without the transverse field term.
Eyal Lubetzky - One of the best experts on this subject based on the ideXlab platform.
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Universality of cutoff for the Ising Model
Annals of Probability, 2017Co-Authors: Eyal LubetzkyAbstract:On any locally-finite geometry, the stochastic Ising Model is known to be contractive when the inverse-temperature ββ is small enough, via classical results of Dobrushin and of Holley in the 1970s. By a general principle proposed by Peres, the dynamics is then expected to exhibit cutoff. However, so far cutoff for the Ising Model has been confirmed mainly for lattices, heavily relying on amenability and log Sobolev inequalities. Without these, cutoff was unknown at any fixed β>0β>0, no matter how small, even in basic examples such as the Ising Model on a binary tree or a random regular graph. We use the new framework of information percolation to show that, in any geometry, there is cutoff for the Ising Model at high enough temperatures. Precisely, on any sequence of graphs with maximum degree dd, the Ising Model has cutoff provided that β
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universality of cutoff for the Ising Model
Annals of Probability, 2017Co-Authors: Eyal LubetzkyAbstract:On any locally-finite geometry, the stochastic Ising Model is known to be contractive when the inverse-temperature ββ is small enough, via classical results of Dobrushin and of Holley in the 1970s. By a general principle proposed by Peres, the dynamics is then expected to exhibit cutoff. However, so far cutoff for the Ising Model has been confirmed mainly for lattices, heavily relying on amenability and log Sobolev inequalities. Without these, cutoff was unknown at any fixed β>0β>0, no matter how small, even in basic examples such as the Ising Model on a binary tree or a random regular graph. We use the new framework of information percolation to show that, in any geometry, there is cutoff for the Ising Model at high enough temperatures. Precisely, on any sequence of graphs with maximum degree dd, the Ising Model has cutoff provided that β<κ/dβ<κ/d for some absolute constant κκ (a result which, up to the value of κκ, is best possible). Moreover, the cutoff location is established as the time at which the sum of squared magnetizations drops to 1, and the cutoff window is O(1)O(1), just as when β=0β=0. Finally, the mixing time from almost every initial state is not more than a factor of 1+eβ1+eβ faster then the worst one (with eβ→0eβ→0 as β→0β→0), whereas the uniform starting state is at least 2−eβ2−eβ times faster.
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information percolation and cutoff for the stochastic Ising Model
Journal of the American Mathematical Society, 2015Co-Authors: Eyal LubetzkyAbstract:We introduce a new framework for analyzing Glauber dynamics for the Ising Model. The traditional approach for obtaining sharp mixing results has been to appeal to estimates on spatial properties of the stationary measure from within a multi-scale analysis of the dynamics. Here we propose to study these simultaneously by examining “information percolation” clusters in the space-time slab. Using this framework, we obtain new results for the Ising Model on (Z/nZ) throughout the high temperature regime: total-variation mixing exhibits cutoff with an O(1)-window around the time at which the magnetization is the square-root of the volume. (Previously, cutoff in the full high temperature regime was only known in dimensions d ≤ 2, and only with an O(log log n)-window.) Furthermore, the new framework opens the door to understanding the effect of the initial state on the mixing time. We demonstrate this on the 1d Ising Model, showing that starting from the uniform (“disordered”) initial distribution asymptotically halves the mixing time, whereas almost every deterministic starting state is asymptotically as bad as starting from the (“ordered”) all-plus state.
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Universality of cutoff for the Ising Model
arXiv: Probability, 2014Co-Authors: Eyal LubetzkyAbstract:On any locally-finite geometry, the stochastic Ising Model is known to be contractive when the inverse-temperature $\beta$ is small enough, via classical results of Dobrushin and of Holley in the 1970's. By a general principle proposed by Peres, the dynamics is then expected to exhibit cutoff. However, so far cutoff for the Ising Model has been confirmed mainly for lattices, heavily relying on amenability and log Sobolev inequalities. Without these, cutoff was unknown at any fixed $\beta>0$, no matter how small, even in basic examples such as the Ising Model on a binary tree or a random regular graph. We use the new framework of information percolation to show that, in any geometry, there is cutoff for the Ising Model at high enough temperatures. Precisely, on any sequence of graphs with maximum degree $d$, the Ising Model has cutoff provided that $\beta
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universality of cutoff for the Ising Model
arXiv: Probability, 2014Co-Authors: Eyal LubetzkyAbstract:On any locally-finite geometry, the stochastic Ising Model is known to be contractive when the inverse-temperature $\beta$ is small enough, via classical results of Dobrushin and of Holley in the 1970's. By a general principle proposed by Peres, the dynamics is then expected to exhibit cutoff. However, so far cutoff for the Ising Model has been confirmed mainly for lattices, heavily relying on amenability and log Sobolev inequalities. Without these, cutoff was unknown at any fixed $\beta>0$, no matter how small, even in basic examples such as the Ising Model on a binary tree or a random regular graph. We use the new framework of information percolation to show that, in any geometry, there is cutoff for the Ising Model at high enough temperatures. Precisely, on any sequence of graphs with maximum degree $d$, the Ising Model has cutoff provided that $\beta<\kappa/d$ for some absolute constant $\kappa$ (a result which, up to the value of $\kappa$, is best possible). Moreover, the cutoff location is established as the time at which the sum of squared magnetizations drops to 1, and the cutoff window is $O(1)$, just as when $\beta=0$. Finally, the mixing time from almost every initial state is not more than a factor of $1+\epsilon_\beta$ faster then the worst one (with $\epsilon_\beta\to0$ as $\beta\to 0$), whereas the uniform starting state is at least $2-\epsilon_\beta$ times faster.
Miguel F Paulos - One of the best experts on this subject based on the ideXlab platform.
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conformal invariance in the long range Ising Model
Nuclear Physics, 2016Co-Authors: Miguel F Paulos, Slava Rychkov, Balt C Van ReesAbstract:We consider the question of conformal invariance of the long-range Ising Model at the critical point. The continuum description is given in terms of a nonlocal field theory, and the absence of a stress tensor invalidates all of the standard arguments for the enhancement of scale invariance to conformal invariance. We however show that several correlation functions, computed to second order in the epsilon expansion, are nontrivially consistent with conformal invariance. We proceed to give a proof of conformal invariance to all orders in the epsilon expansion, based on the description of the long-range Ising Model as a defect theory in an auxiliary higher-dimensional space. A detailed review of conformal invariance in the d-dimensional short-range Ising Model is also included and may be of independent interest.
