Uniqueness of Quantum Markov Chain Associated with XY-Ising Model on Cayley Tree of Order Two

2017 ◽  
Vol 24 (02) ◽  
pp. 1750010 ◽  
Author(s):  
Farrukh Mukhamedov ◽  
Soueidy El Gheteb
Keyword(s):  
Markov Chain ◽  
Ising Model ◽  
Markov Chains ◽  
Boundary Conditions ◽  
Finite Volume ◽  
Cayley Tree ◽  
Limit State ◽  
Weak Limit ◽  
Quantum Markov Chain ◽  
Quantum Markov Chains

In this paper, we consider backward and forward Quantum Markov Chains (QMC) associated with XY -Ising model on the Cayley tree of order two. We construct finite volume states with boundary conditions, and define QMC as a weak limit of those states which depend on the boundary conditions. We prove that the limit state is a unique QMC associated with such a model, this means the QMC does not depend on the boundary conditions. Moreover, we observe the relation between backward and forward QMC.

scholarly journals ON QUANTUM MARKOV CHAINS ON CAYLEY TREE I: UNIQUENESS OF THE ASSOCIATED CHAIN WITH XY-MODEL ON THE CAYLEY TREE OF ORDER TWO

2011 ◽  
Vol 14 (03) ◽  
pp. 443-463 ◽  
Author(s):  
LUIGI ACCARDI ◽  
FARRUKH MUKHAMEDOV ◽  
MANSOOR SABUROV
Keyword(s):  
Markov Chains ◽  
Boundary Conditions ◽  
Cayley Tree ◽  
Weak Limit ◽  
Finite Volumes ◽  
Xy Model ◽  
Quantum Markov Chains

In this paper we study forward quantum Markov chains (QMC) defined on Cayley tree. A construction of such QMC is provided, namely we construct states on finite volumes with boundary conditions, and define QMC as a weak limit of those states which depends on the boundary conditions. Using the provided construction, we investigate QMC associated with XY-model on a Cayley tree of order two. We prove uniqueness of QMC associated with such a model, this means the QMC does not depend on the boundary conditions.

scholarly journals On Quantum Markov Chains on Cayley Tree III: Ising Model

2014 ◽  
Vol 157 (2) ◽  
pp. 303-329 ◽  
Author(s):  
Luigi Accardi ◽  
Farrukh Mukhamedov ◽  
Mansoor Saburov
Keyword(s):  
Ising Model ◽  
Markov Chains ◽  
Cayley Tree ◽  
Quantum Markov Chains

scholarly journals A quantum Markov chain approach to phase transitions for quantum Ising model with competing XY-interactions on a Cayley tree

2020 ◽  
Vol 61 (9) ◽  
pp. 093505
Author(s):  
Farrukh Mukhamedov ◽  
Abdessatar Barhoumi ◽  
Abdessatar Souissi ◽  
Soueidy El Gheteb
Keyword(s):  
Markov Chain ◽  
Phase Transitions ◽  
Ising Model ◽  
Cayley Tree ◽  
Quantum Markov Chain ◽  
Markov Chain Approach ◽  
Quantum Ising Model

scholarly journals Quantum Markov chains and classical random sequences

1995 ◽  
Vol 139 ◽  
pp. 173-183 ◽  
Author(s):  
Yun-Gang Lu
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Valence Bond ◽  
Natural Generalization ◽  
Random Sequences ◽  
Quantum Markov Chain ◽  
Quantum Markov Chains

Quantum Markov chain introduced by Accardi (cf. [1,2,3]) is one of natural generalization of classical Markov chain. It has many interesting applications in physics and the most important one is given by the paper of Fannes-Nachtergaele-Werner ([4]), where an application of quantum Markov chain’s technique enables us to understand the Valence bond states well.

scholarly journals Markov Chains Conditioned Never to Wait Too Long at the Origin

2009 ◽  
Vol 46 (03) ◽  
pp. 812-826
Author(s):  
Saul Jacka
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Sufficient Conditions ◽  
Weak Limit ◽  
The State ◽  
Coin Tossing ◽  
First Time ◽  
Vague Convergence ◽  
Irreducible Markov Chain

Motivated by Feller's coin-tossing problem, we consider the problem of conditioning an irreducible Markov chain never to wait too long at 0. Denoting by τ the first time that the chain,X, waits for at least one unit of time at the origin, we consider conditioning the chain on the event (τ›T). We show that there is a weak limit asT→∞ in the cases where either the state space is finite orXis transient. We give sufficient conditions for the existence of a weak limit in other cases and show that we have vague convergence to a defective limit if the time to hit zero has a lighter tail than τ and τ is subexponential.

scholarly journals Uniqueness of quantum Markov chains associated with an XY-model on a cayley tree of order 2

2011 ◽  
Vol 90 (1-2) ◽  
pp. 162-174 ◽  
Author(s):  
L. Accardi ◽  
F. M. Mukhamedov ◽  
M. Kh. Saburov
Keyword(s):  
Markov Chains ◽  
Cayley Tree ◽  
Xy Model ◽  
Quantum Markov Chains
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