Alphabeta MathPolymer analyticity and dilute continuum correlations
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1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Areas of Elementary Plane Figures
- Binary Operations, Monoids, Groups and Subgroups
- Cayley Graphs, Word Metrics and Quasi-Isometry
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Complex Differentiability and the Cauchy–Riemann Equations
- Complex Norms and Realification
- Complex Power Series and Analytic Functions
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Countability Axioms and Cardinal Functions
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Ensembles And Limits
- Filters and Ultrafilters
- Finite Counting, Factorials and Binomial Coefficients
- Finite-range lattice interaction and conditional energy
- Foundations of the Real Numbers for Analysis
- Graphs, Walks and Connectivity
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Lebesgue Measure on Euclidean Space
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Metric Spaces
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Normal Subgroups and Quotient Groups
- Order, Zorn's Lemma, and the Axiom of Choice
- Outer Measure and the Caratheodory Extension Theorem
- Pointwise Extended-Real Function Products
- Polynomial Rings, the Division Algorithm and Roots
- Power Series and Real-Analytic Functions
- Product Measures and the Fubini Tonelli Theorems
- Properties of the Integral and the Working FTC
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sigma Algebras and Borel Sets
- Simple Field Extensions and the Construction of the Complex Numbers
- Subspaces, Products, and Quotients
- Suprema and Infima
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Lebesgue Integral and the Convergence Theorems
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Topology of Euclidean Space
- The ZFC Axioms and the Basic Set Constructions
- Topological Spaces and Continuity
- Topology of ℝ
- Trees, Forests and Spanning Trees
- Vector Spaces, Linear Subspaces, Span and Direct Sums
2 · Logical flowchart
3 · Items
Definition
Abstract polymer model and connected coefficients
Lemma
Rooted connected polymer sums obey the tree majorant
Theorem
A strict polymer majorant gives a zero-free finite partition function
Lemma
Stable continuum Mayer coefficients obey an integrated tree bound
Theorem
The stable continuum Mayer pressure series converges in an explicit dilute disk
Sources
- Sacha Friedli and Yvan Velenik, Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction, Chapter 5, Cluster Expansion
- Daniel Ueltschi, Cluster expansions and correlation functions
- Suren Poghosyan and Daniel Ueltschi, Abstract cluster expansion with applications to statistical mechanical systems
- Aldo Procacci and Sergio A. Yuhjtman, Convergence of Mayer and Virial expansions and the Penrose tree-graph identity