scholarly journals Estimating the spectral gap of a trace-class Markov operator

2019 ◽  
Vol 13 (1) ◽  
pp. 1790-1822
Author(s):  
Qian Qin ◽  
James P. Hobert ◽  
Kshitij Khare
Keyword(s):  
Spectral Gap ◽  
Markov Operator ◽  
Trace Class

scholarly journals Spectral Theory for Weakly Reversible Markov Chains

2012 ◽  
Vol 49 (01) ◽  
pp. 245-265 ◽  
Author(s):  
Achim Wübker
Keyword(s):  
Markov Chains ◽  
Spectral Gap ◽  
Transition Probabilities ◽  
Markov Operator ◽  
Spectral Gaps ◽  
Reversible Markov Chains ◽  
General State Space ◽  
Stationary Transition ◽  
Positive Recurrent ◽  
Necessary And Sufficient

The theory of L 2-spectral gaps for reversible Markov chains has been studied by many authors. In this paper we consider positive recurrent general state space Markov chains with stationary transition probabilities. Replacing the assumption of reversibility with a weaker assumption, we still obtain a simple necessary and sufficient condition for the spectral gap property of the associated Markov operator in terms of the isoperimetric constant. We show that this result can be applied to a large class of Markov chains, including those that are related to positive recurrent finite-range random walks on Z.

scholarly journals Spectral Theory for Weakly Reversible Markov Chains

2012 ◽  
Vol 49 (1) ◽  
pp. 245-265 ◽  
Author(s):  
Achim Wübker
Keyword(s):  
Markov Chains ◽  
Spectral Gap ◽  
Transition Probabilities ◽  
Markov Operator ◽  
Spectral Gaps ◽  
Reversible Markov Chains ◽  
General State Space ◽  
Stationary Transition ◽  
Positive Recurrent ◽  
Necessary And Sufficient

The theory of L2-spectral gaps for reversible Markov chains has been studied by many authors. In this paper we consider positive recurrent general state space Markov chains with stationary transition probabilities. Replacing the assumption of reversibility with a weaker assumption, we still obtain a simple necessary and sufficient condition for the spectral gap property of the associated Markov operator in terms of the isoperimetric constant. We show that this result can be applied to a large class of Markov chains, including those that are related to positive recurrent finite-range random walks on Z.

A Note on Fitting Markov Operator Credit Risk Models

2008 ◽  
Author(s):  
Harley Thompson ◽  
Jonathan Harris
Keyword(s):  
Credit Risk ◽  
Markov Operator ◽  
Risk Models

scholarly journals Quantum Orlicz Spaces in Information Geometry

2004 ◽  
Vol 11 (04) ◽  
pp. 359-375 ◽  
Author(s):  
R. F. Streater
Keyword(s):  
Quantum Information ◽  
Tangent Space ◽  
Selfadjoint Operator ◽  
Affine Connection ◽  
Orlicz Spaces ◽  
Trace Class ◽  
Information Geometry ◽  
Young Function ◽  
Affine Structure ◽  
Cotangent Space

Let H0 be a selfadjoint operator such that Tr e−βH0 is of trace class for some β < 1, and let χɛ denote the set of ɛ-bounded forms, i.e., ∥(H0+C)−1/2−ɛX(H0+C)−1/2+ɛ∥ < C for some C > 0. Let χ := Span ∪ɛ∈(0,1/2]χɛ. Let [Formula: see text] denote the underlying set of the quantum information manifold of states of the form ρx = e−H0−X−ψx, X ∈ χ. We show that if Tr e−H0 = 1. 1. the map Φ, [Formula: see text] is a quantum Young function defined on χ 2. The Orlicz space defined by Φ is the tangent space of [Formula: see text] at ρ0; its affine structure is defined by the (+1)-connection of Amari 3. The subset of a ‘hood of ρ0, consisting of p-nearby states (those [Formula: see text] obeying C−1ρ1+p ≤ σ ≤ Cρ1 − p for some C > 1) admits a flat affine connection known as the (−1) connection, and the span of this set is part of the cotangent space of [Formula: see text] 4. These dual structures extend to the completions in the Luxemburg norms.

scholarly journals Spectral Gap of the Largest Eigenvalue of the Normalized Graph Laplacian

2021 ◽  
Author(s):  
Jürgen Jost ◽  
Raffaella Mulas ◽  
Florentin Münch
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Spectral Gap ◽  
Complete Bipartite Graph ◽  
Graph Laplacian ◽  
New Method ◽  
Single Edge ◽  
Largest Eigenvalue ◽  
Complement Graph ◽  
The Largest Eigenvalue

AbstractWe offer a new method for proving that the maxima eigenvalue of the normalized graph Laplacian of a graph with n vertices is at least $$\frac{n+1}{n-1}$$ n + 1 n - 1 provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size $$\frac{n-1}{2}$$ n - 1 2 . With the same method, we also prove a new lower bound to the largest eigenvalue in terms of the minimum vertex degree, provided this is at most $$\frac{n-1}{2}$$ n - 1 2 .

scholarly journals Free partition functions and an averaged holographic duality

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Nima Afkhami-Jeddi ◽  
Henry Cohn ◽  
Thomas Hartman ◽  
Amirhossein Tajdini
Keyword(s):  
Partition Function ◽  
Spectral Gap ◽  
Central Charge ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Partition Functions ◽  
General Constraints ◽  
Dimensional Gravity ◽  
Holographic Duality

Abstract We study the torus partition functions of free bosonic CFTs in two dimensions. Integrating over Narain moduli defines an ensemble-averaged free CFT. We calculate the averaged partition function and show that it can be reinterpreted as a sum over topologies in three dimensions. This result leads us to conjecture that an averaged free CFT in two dimensions is holographically dual to an exotic theory of three-dimensional gravity with U(1)c×U(1)c symmetry and a composite boundary graviton. Additionally, for small central charge c, we obtain general constraints on the spectral gap of free CFTs using the spinning modular bootstrap, construct examples of Narain compactifications with a large gap, and find an analytic bootstrap functional corresponding to a single self-dual boson.

scholarly journals Persistence of the spectral gap for the Landau–Pekar equations

2021 ◽  
Vol 111 (1) ◽  
Author(s):  
Dario Feliciangeli ◽  
Simone Rademacher ◽  
Robert Seiringer
Keyword(s):  
Initial Data ◽  
Spectral Gap ◽  
Effective Hamiltonian ◽  
Adiabatic Theorem ◽  
Strongly Coupled

AbstractThe Landau–Pekar equations describe the dynamics of a strongly coupled polaron. Here, we provide a class of initial data for which the associated effective Hamiltonian has a uniform spectral gap for all times. For such initial data, this allows us to extend the results on the adiabatic theorem for the Landau–Pekar equations and their derivation from the Fröhlich model obtained in previous works to larger times.

scholarly journals Nonzero spectral gap in several uniformly spin-2 and hybrid spin-1 and spin-2 AKLT models

2021 ◽  
Vol 3 (1) ◽  
Author(s):  
Wenhan Guo ◽  
Nicholas Pomata ◽  
Tzu-Chieh Wei
Keyword(s):  
Spectral Gap

scholarly journals Spectral Gap of the Antiferromagnetic Lipkin–Meshkov–Glick Model

2018 ◽  
Vol 195 (2) ◽  
pp. 718-728
Author(s):  
R. G. Unanyan
Keyword(s):  
Spectral Gap

Homogenization of the parabolic equation with periodic coefficients at the edge of a spectral gap

2021 ◽  
pp. 1-33
Author(s):  
A. R. Akhmatova ◽  
E. S. Aksenova ◽  
V. A. Sloushch ◽  
T. A. Suslina
Keyword(s):  
Parabolic Equation ◽  
Spectral Gap ◽  
Periodic Coefficients
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