preface
comments on the use of this book
part i: large deviations and statistical mechanics
chapter i. introduction to large deviations
i.1. overview
i.2. large deviations for 1.i.d. random variables with a finite state space
i.3. levels-1 and 2 for coin tossing
i.4. levels-1 and 2 for i.i.d. random variables with a finite state space
i.5. level-3: empirical pair measure
i.6. level-3: empirical process
i.7. notes
i.8. problems
chapter ii. large deviation property and asymptotics of integrals
ii.1. introduction
ii.2. levels-l, 2, and 3 large deviations for i.i.d. random vectors
ii.3. the definition of large deviation property
ii.4. statement of large deviation properties for levels-l, 2, and 3
ii.5. contraction principles
ii.6. large deviation property for random vectors and exponential convergence
ii.7. varadhan﹀s theorem on the asymptotics of integrals
ii.8. notes
ii.9. problems
chapter iii. large deviations and the discrete ideal gas
iii.1. introduction
iii.2. physics prelude: thermodynamics
iii.3. the discrete ideal gas and the microcanonical ensemble
iii.4. thermodynamic limit, exponential convergence, and equilibrium values
iii.5. the maxweli-boltzmann distribution and temperature
iii.6. the canonical ensemble and its equivalence with the microcanonical ensemble
iii.7. a derivation of a thermodynamic equation
ill.8. the gibbs variational formula and principle
iii.9. notes
iii. 10. problems
chapter iv. ferromagnetic models on z
iv.1. introduction
iv.2. an overview of ferromagnetic models
iv.3. finite-volume gibbs states on 77
iv.4. spontaneous magnetization for the curie-weiss model
iv.5. spontaneous magnetization for general ferromagnets on
iv.6. infinite-volume gibbs states and phase transitions
iv.7. the gibbs variational formula and principle
iv.8. notes
iv.9. problems
chapter v. magnetic models on 7/d and on the circle
v.1. introduction
v.2. finite-volume gibbs states on zd, d ≧ 1
v.3. moment inequalities
v.4. properties of the magnetization and the gibbs free energy
v.5. spontaneous magnetization on z, d ≧ 2, via the peierls argument
v.6. infinite-volume gibbs states and phase transitions
v.7. infinite-volume gibbs states and the central limit theorem
v.8. critical phenomena and the breakdown of the central limit theorem
v.9. three faces of the curie-weiss model
v. 10. the circle model and random waves
v.11. a postscript on magnetic models
v.12. notes
v.13. problems
part ii: convexity and proofs of large deviation theorems
chapter vi. convex functions and the legendre-fenchel transform
vii.1. introduction
vi.2. basic definitions
vi.3. properties of convex functions
vi.4. a one-dimensional example pf the legendre-fenchel transform
vi.5. the legendre-fenchel transform for convex functions on ra
vi.6. notes
vi.7. problems
chapter vii. large deviations for random vectors
vii. i. statement of results
vii.2. properties of i
vii.3. proof of the large deviation bounds for d = 1
vii.4. proof of the large deviation bounds for d≧ 1
vii.5. level-i large deviations for i.i.d. random vectors
vii.6. exponential convergence and proof of theorem ii.6.3
vii.7. notes
vii.8. problems
chapter viii. level-2 large deviations for i.i.d. random vectors
viii. 1. introduction
viii.2. the level-2 large deviation theorem
viii.3. the contraction principle relating levels-i and 2 (d = 1)
viii.4. the contraction principle relating levels-1 and 2 (d ≧ 2)
viii.5. notes
viii.6. problems
chapter ix. level-3 large deviations for i.i.d. random vectors
ix. 1. statement of results
ix.2. properties of the level-3 entropy function
ix.3. contraction principles
ix.4. proof of the level-3 large deviation bounds
ix.5. notes
ix.6. problems
appendices
appendix a: probability
a.1. introduction
a.2. measurability
a.3. product spaces
a.4. probability measures and expectation
a.5. convergence of random vectors
a.6. conditional expectation, conditional probability, and regular conditional distribution
a.7. the koimogorov existence theorem
a.8. weak convergence of probability measures on a metric space
a.9. the space ms((rd)z) and the ergodic theorem
a.10. n-dependent markov chains
a.11. probability measures on the space { 1, - 1}zd
appendix b: proofs of two theorems in section ii.7
b.i. proof of theorem ii.7.1
b.2. proof of theorem ii.7.2
appendix c: equivalent notions of infinite-volume measures for spin systems
c.i. introduction
c.2. two-body interactions and infinite-volume gibbs states
c.3. many-body interactions and infinite-volume gibbs states
c.4. dlr states
c.5. the gibbs variational formula and principle
c.6. solution of the gibbs variational formula for finite-range interactions on z
appendix d: existence of the specific gibbs free energy
d.1. existence along hypercubes
d.2. an extension
list of frequently used symbols
references
author index
subject index
