scholarly journals Bounding the Partition Function of Spin-Systems

10.37236/1098 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
David J. Galvin
Keyword(s):  
Ising Model ◽  
Partition Function ◽  
Probability Distribution ◽  
Statistical Physics ◽  
Hard Core ◽  
Spin Systems ◽  
Normalizing Constant ◽  
Graph Homomorphisms ◽  
Hard Core Model ◽  
Regular Bipartite Graph

With a graph $G=(V,E)$ we associate a collection of non-negative real weights $\bigcup_{v\in V}\{\lambda_{i,v}:1\leq i \leq m\} \cup \bigcup_{uv \in E} \{\lambda_{ij,uv}:1\leq i \leq j \leq m\}.$ We consider the probability distribution on $\{f:V\rightarrow\{1,\ldots,m\}\}$ in which each $f$ occurs with probability proportional to $\prod_{v \in V}\lambda_{f(v),v}\prod_{uv \in E}\lambda_{f(u)f(v),uv}$. Many well-known statistical physics models, including the Ising model with an external field and the hard-core model with non-uniform activities, can be framed as such a distribution. We obtain an upper bound, independent of $G$, for the partition function (the normalizing constant which turns the assignment of weights on $\{f:V\rightarrow\{1,\ldots,m\}\}$ into a probability distribution) in the case when $G$ is a regular bipartite graph. This generalizes a bound obtained by Galvin and Tetali who considered the simpler weight collection $\{\lambda_i:1 \leq i \leq m\} \cup \{\lambda_{ij}:1 \leq i \leq j \leq m\}$ with each $\lambda_{ij}$ either $0$ or $1$ and with each $f$ chosen with probability proportional to $\prod_{v \in V}\lambda_{f(v)}\prod_{uv \in E}\lambda_{f(u)f(v)}$. Our main tools are a generalization to list homomorphisms of a result of Galvin and Tetali on graph homomorphisms and a straightforward second-moment computation.

scholarly journals Inapproximability of the Partition Function for the Antiferromagnetic Ising and Hard-Core Models

2016 ◽  
Vol 25 (4) ◽  
pp. 500-559 ◽  
Author(s):  
ANDREAS GALANIS ◽  
DANIEL ŠTEFANKOVIČ ◽  
ERIC VIGODA
Keyword(s):  
Ising Model ◽  
Partition Function ◽  
Maximum Degree ◽  
Hard Core ◽  
Regular Tree ◽  
Infinite Tree ◽  
Critical Activity ◽  
Antiferromagnetic Ising Model ◽  
Hard Core Model ◽  
Constant Maximum

Recent inapproximability results of Sly (2010), together with an approximation algorithm presented by Weitz (2006), establish a beautiful picture of the computational complexity of approximating the partition function of the hard-core model. Let λc($\mathbb{T}_{\Delta}$) denote the critical activity for the hard-model on the infinite Δ-regular tree. Weitz presented anFPTASfor the partition function when λ < λc($\mathbb{T}_{\Delta}$) for graphs with constant maximum degree Δ. In contrast, Sly showed that for all Δ ⩾ 3, there exists εΔ> 0 such that (unless RP = NP) there is noFPRASfor approximating the partition function on graphs of maximum degree Δ for activities λ satisfying λc($\mathbb{T}_{\Delta}$) < λ < λc($\mathbb{T}_{\Delta}$) + εΔ.We prove that a similar phenomenon holds for the antiferromagnetic Ising model. Sinclair, Srivastava and Thurley (2014) extended Weitz's approach to the antiferromagnetic Ising model, yielding anFPTASfor the partition function for all graphs of constant maximum degree Δ when the parameters of the model lie in the uniqueness region of the infinite Δ-regular tree. We prove the complementary result for the antiferromagnetic Ising model without external field, namely, that unless RP = NP, for all Δ ⩾ 3, there is noFPRASfor approximating the partition function on graphs of maximum degree Δ when the inverse temperature lies in the non-uniqueness region of the infinite tree$\mathbb{T}_{\Delta}$. Our proof works by relating certain second moment calculations for random Δ-regular bipartite graphs to the tree recursions used to establish the critical points on the infinite tree.

An Entropy Approach to the Hard-Core Model on Bipartite Graphs

2001 ◽  
Vol 10 (3) ◽  
pp. 219-237 ◽  
Author(s):  
JEFF KAHN
Keyword(s):  
Phase Transition ◽  
Bipartite Graph ◽  
Upper Bound ◽  
Hard Core ◽  
Independent Set ◽  
Independent Sets ◽  
Bipartite Graphs ◽  
Core Model ◽  
Hard Core Model ◽  
Regular Bipartite Graph

We use entropy ideas to study hard-core distributions on the independent sets of a finite, regular bipartite graph, specifically distributions according to which each independent set I is chosen with probability proportional to λ[mid ]I[mid ] for some fixed λ > 0. Among the results obtained are rather precise bounds on occupation probabilities; a ‘phase transition’ statement for Hamming cubes; and an exact upper bound on the number of independent sets in an n-regular bipartite graph on a given number of vertices.

scholarly journals Weighted counting of solutions to sparse systems of equations

2019 ◽  
Vol 28 (5) ◽  
pp. 696-719 ◽  
Author(s):  
Alexander Barvinok ◽  
Guus Regts
Keyword(s):  
Linear Codes ◽  
Linear Equations ◽  
Hard Core ◽  
Weighted Graph ◽  
Partition Functions ◽  
Weight Enumerators ◽  
Graph Homomorphisms ◽  
Sparse Systems ◽  
Hard Core Model ◽  
Homogeneous Linear

AbstractGiven complex numbers w1,…,wn, we define the weight w(X) of a set X of 0–1 vectors as the sum of $w_1^{x_1} \cdots w_n^{x_n}$ over all vectors (x1,…,xn) in X. We present an algorithm which, for a set X defined by a system of homogeneous linear equations with at most r variables per equation and at most c equations per variable, computes w(X) within relative error ∊ > 0 in (rc)O(lnn-ln∊) time provided $|w_j| \leq \beta (r \sqrt{c})^{-1}$ for an absolute constant β > 0 and all j = 1,…,n. A similar algorithm is constructed for computing the weight of a linear code over ${\mathbb F}_p$. Applications include counting weighted perfect matchings in hypergraphs, counting weighted graph homomorphisms, computing weight enumerators of linear codes with sparse code generating matrices, and computing the partition functions of the ferromagnetic Potts model at low temperatures and of the hard-core model at high fugacity on biregular bipartite graphs.

scholarly journals Reconstruction Thresholds on Regular Trees

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
James B. Martin
Keyword(s):  
Statistical Physics ◽  
Gibbs Measures ◽  
Rooted Tree ◽  
Hard Core ◽  
Core Model ◽  
A Value ◽  
Binary State ◽  
International Audience ◽  
Hard Core Model ◽  
Special Case

International audience We consider themodel of broadcasting on a tree, with binary state space, on theinfinite rooted tree $T^k$ in which each node has $k$ children. The root of the tree takesa random value $0$ or $1$, and then each node passes a value independently to each of its children according to a $2x2$ transition matrix $\mathbf{P}$. We say that reconstruction is possible if the values at the dth level of the tree contain non-vanishing information about the value at the root as $d→∞$. Extending a method of Brightwell and Winkler, we obtain new conditions under which reconstruction is impossible, both in the general case and in the special case $p_11=0$. The latter case is closely related to the hard-core model from statistical physics; a corollary of our results is that, for the hard-core model on the $(k+1)$-regular tree with activity $λ =1$, the unique simple invariant Gibbs measure is extremal in the set of Gibbs measures, for any $k ≥ 2$.

scholarly journals Finding the dominant zero of the energy probability distribution

2021 ◽  
pp. 2150155
Author(s):  
J. J. Carvalho ◽  
A. L. Mota
Keyword(s):  
Monte Carlo ◽  
Critical Temperature ◽  
Ising Model ◽  
Partition Function ◽  
Probability Distribution ◽  
Processing Time ◽  
Computational Procedure ◽  
Thermodynamic System ◽  
Computer Processing ◽  
Square Lattice

In this work, we present a computational procedure to locate the dominant Fisher zero of the partition function of a thermodynamic system. The procedure greatly reduces the required computer processing time to find the dominant zero when compared to other dominant zero search procedures. As a consequence, when the partition function results in very large polynomials, the accuracy of the results can be increased, since less drastic truncation of the polynomials (or even no truncation) is necessary. We apply the procedure to the 2D Ising model in a square lattice, obtaining very accurate results for the critical temperature and some of the critical exponents of the model. We also show the results obtained when the technique is used with the Monte Carlo simulated 2D Ising model in large lattices.

scholarly journals Statistical Thermodynamics of Economic Systems

2011 ◽  
Vol 2011 ◽  
pp. 1-8 ◽  
Author(s):  
Hernando Quevedo ◽  
María N. Quevedo
Keyword(s):  
Phase Transitions ◽  
Partition Function ◽  
Probability Distribution ◽  
Statistical Physics ◽  
Mean Value ◽  
Economic Systems ◽  
Macroeconomic Variables ◽  
Multiplicative Systems ◽  
The Mean ◽  
Linear And Nonlinear

We formulate the thermodynamics of economic systems in terms of an arbitrary probability distribution for a conserved economic quantity. As in statistical physics, thermodynamic macroeconomic variables emerge as the mean value of microeconomic variables, and their determination is reduced to the computation of the partition function, starting from an arbitrary function. Explicit hypothetical examples are given which include linear and nonlinear economic systems as well as multiplicative systems such as those dominated by a Pareto law distribution. It is shown that the macroeconomic variables can be drastically changed by choosing the microeconomic variables in an appropriate manner. We propose to use the formalism of phase transitions to study severe changes of macroeconomic variables.

scholarly journals Independent sets of a given size and structure in the hypercube

2022 ◽  
pp. 1-19
Author(s):  
Matthew Jenssen ◽  
Will Perkins ◽  
Aditya Potukuchi
Keyword(s):  
Central Limit ◽  
Statistical Physics ◽  
Limit Theorems ◽  
Hard Core ◽  
Independent Sets ◽  
Structural Features ◽  
Combinatorial Enumeration ◽  
Polymer Models ◽  
Local Central Limit Theorem ◽  
Hard Core Model

Abstract We determine the asymptotics of the number of independent sets of size $\lfloor \beta 2^{d-1} \rfloor$ in the discrete hypercube $Q_d = \{0,1\}^d$ for any fixed $\beta \in (0,1)$ as $d \to \infty$ , extending a result of Galvin for $\beta \in (1-1/\sqrt{2},1)$ . Moreover, we prove a multivariate local central limit theorem for structural features of independent sets in $Q_d$ drawn according to the hard-core model at any fixed fugacity $\lambda>0$ . In proving these results we develop several general tools for performing combinatorial enumeration using polymer models and the cluster expansion from statistical physics along with local central limit theorems.

scholarly journals The bulk, surface and corner free energies of the anisotropic triangular Ising model

2020 ◽  
Vol 476 (2234) ◽  
pp. 20190713
Author(s):  
Rodney J. Baxter
Keyword(s):  
Free Energy ◽  
Ising Model ◽  
Partition Function ◽  
Temperature Series ◽  
Isotropic Case ◽  
Series Expansions ◽  
Exact Results ◽  
Free Energies ◽  
Bulk Surface ◽  
Bulk Free Energy

We consider the anisotropic Ising model on the triangular lattice with finite boundaries, and use Kaufman’s spinor method to calculate low-temperature series expansions for the partition function to high order. From these, we can obtain 108-term series expansions for the bulk, surface and corner free energies. We extrapolate these to all terms and thereby conjecture the exact results for each. Our results agree with the exactly known bulk-free energy and with Cardy and Peschel’s conformal invariance predictions for the dominant behaviour at criticality. For the isotropic case, they also agree with Vernier and Jacobsen’s conjecture for the 60 ° corners.

Sign in / Sign up

Export Citation Format

Share Document