Markov Chain Monte Carlo Methods in Quantum Field Theories: A Modern Primer (SpringerBriefs in Physics)

Foreword
Preface
Acknowledgements
Contents
1 Monte Carlo Method for Integration
	1.1 Numerical Integration
	1.2 Composite Formulas for Numerical Integration
		1.2.1 Composite Rectangle Rule
		1.2.2 Composite Midpoint Rule
		1.2.3 Composite Trapezoidal Rule
		1.2.4 Composite Simpson\'s Rule
	1.3 Random Numbers
		1.3.1 Physical Random Numbers
		1.3.2 Pseudo-random Numbers
		1.3.3 Random Numbers Using UNIX Function Drand48()
		1.3.4 Random Numbers Using a Seed
		1.3.5 Random Numbers from Non-uniform Distributions
	1.4 Monte Carlo Method
		1.4.1 Worked Example—Composite Midpoint Rule
		1.4.2 Worked Example—Composite Simpson\'s Rule
		1.4.3 Worked Example—Monte Carlo Integration
	1.5 Error in Monte Carlo Integration
	1.6 When is Monte Carlo Good for Integration?
	1.7 When does Monte Carlo Fail?
2 Monte Carlo with Importance Sampling
	2.1 Naive Sampling and Importance Sampling
	2.2 Worked Example—Importance Sampling
	2.3 When does Importance Sampling Fail?
3 Markov Chains
	3.1 Properties of Markov Chains
		3.1.1 Irreducibility
		3.1.2 Aperiodicity
	3.2 Convergence of Markov Chains
4 Markov Chain Monte Carlo
	4.1 Metropolis-Hastings Algorithm
	4.2 Metropolis Algorithm
	4.3 Worked Example—Metropolis for Gaussian Integral
	4.4 Thermalization in Markov Chain Monte Carlo
5 MCMC and Feynman Path Integrals
	5.1 Transition Amplitudes
	5.2 Feynman Path Integrals
	5.3 Worked Example—Supersymmetry Breaking
	5.4 Worked Example—Simple Harmonic Oscillator
	5.5 Worked Example—Unitary Matrix Model
6 Reliability of Simulations
	6.1 Auto-correlation Time
	6.2 Error Analysis
7 Hybrid (Hamiltonian) Monte Carlo
	7.1 Hamilton\'s Equations
	7.2 Properties of Hamiltonian Dynamics
	7.3 Leapfrog Method
	7.4 MCMC from Hamiltonian Dynamics
		7.4.1 Joint Probability Distribution
		7.4.2 Tuning HMC Algorithm
		7.4.3 HMC Algorithm—Step by Step
	7.5 Worked Example—HMC for Gaussian Model
	7.6 Worked Example—HMC for Supersymmetric Model
8 MCMC and Quantum Field Theories  on a Lattice
9 Machine Learning and Quantum Field Theories
Appendix C++ Programs
A.1 Random Numbers from a Uniform Distribution
A.2 Random Numbers with a Seed
A.3 Random Numbers from a Gaussian Distribution
A.4 Numerical Integration—Composite Midpoint Rule
A.5 Numerical Integration—Composite Simpson\'s Rule
A.6 Numerical Integration—Monte Carlo Method
A.7 Numerical Integration—Naive Monte Carlo Sampling
A.8 Numerical Integration—Importance Sampling Monte Carlo
A.9 Metropolis Algorithm for Gaussian Model
A.10 Supersymmetric Model—Metropolis Sampling
A.11 Metropolis for Simple Harmonic Oscillator
A.12 Metropolis for Unitary Matrix Model
A.13 Computing Auto-correlation Time
A.14 Hamiltonian Monte Carlo for Gaussian Model
Appendix  References