scholarly journals Contraction: A Unified Perspective of Correlation Decay and Zero-Freeness of 2-Spin Systems

2021 ◽  
Vol 185 (2) ◽  
Author(s):  
Shuai Shao ◽  
Yuxin Sun
Keyword(s):  
Partition Function ◽  
Approximation Algorithms ◽  
External Field ◽  
Spin System ◽  
Spin Systems ◽  
Interaction Parameters ◽  
Entire Family ◽  
Bounded Degree ◽  
Correlation Decay ◽  
Complex Valued

AbstractWe study the connection between the correlation decay property (more precisely, strong spatial mixing) and the zero-freeness of the partition function of 2-spin systems on graphs of bounded degree. We show that for 2-spin systems on an entire family of graphs of a given bounded degree, the contraction property that ensures correlation decay exists for certain real parameters implies the zero-freeness of the partition function and the existence of correlation decay for some corresponding complex neighborhoods. Based on this connection, we are able to extend any real parameter of which the 2-spin system on graphs of bounded degree exhibits correlation decay to its complex neighborhood where the partition function is zero-free and correlation decay still exists. We give new zero-free regions in which the edge interaction parameters and the uniform external field are all complex-valued, and we show the existence of correlation decay for such complex regions. As a consequence, we obtain approximation algorithms for computing the partition function of 2-spin systems on graphs of bounded degree for these complex parameter settings.

scholarly journals Fast Algorithms for General Spin Systems on Bipartite Expanders

2021 ◽  
Vol 13 (4) ◽  
pp. 1-18
Author(s):  
Andreas Galanis ◽  
Leslie Ann Goldberg ◽  
James Stewart
Keyword(s):  
Partition Function ◽  
Spin System ◽  
Weighted Graph ◽  
Arbitrary Spin ◽  
Spin Assignment ◽  
Spin Systems ◽  
Regular Graphs ◽  
Canonical Class ◽  
Finite Set ◽  
Polymer Models

A spin system is a framework in which the vertices of a graph are assigned spins from a finite set. The interactions between neighbouring spins give rise to weights, so a spin assignment can also be viewed as a weighted graph homomorphism. The problem of approximating the partition function (the aggregate weight of spin assignments) or of sampling from the resulting probability distribution is typically intractable for general graphs. In this work, we consider arbitrary spin systems on bipartite expander Δ-regular graphs, including the canonical class of bipartite random Δ-regular graphs. We develop fast approximate sampling and counting algorithms for general spin systems whenever the degree and the spectral gap of the graph are sufficiently large. Roughly, this guarantees that the spin system is in the so-called low-temperature regime. Our approach generalises the techniques of Jenssen et al. and Chen et al. by showing that typical configurations on bipartite expanders correspond to “bicliques” of the spin system; then, using suitable polymer models, we show how to sample such configurations and approximate the partition function in Õ( n 2 ) time, where n is the size of the graph.

scholarly journals Approximation Algorithms for Complex-Valued Ising Models on Bounded Degree Graphs

Quantum ◽  
2019 ◽  
Vol 3 ◽  
pp. 162 ◽  
Author(s):  
Ryan L. Mann ◽  
Michael J. Bremner
Keyword(s):  
Partition Function ◽  
Approximation Scheme ◽  
Ising Models ◽  
Quantum Circuits ◽  
Bounded Degree ◽  
Graph Polynomials ◽  
Bounded Degree Graphs ◽  
Complex Zeros ◽  
Complex Valued ◽  
Output Probability

We study the problem of approximating the Ising model partition function with complex parameters on bounded degree graphs. We establish a deterministic polynomial-time approximation scheme for the partition function when the interactions and external fields are absolutely bounded close to zero. Furthermore, we prove that for this class of Ising models the partition function does not vanish. Our algorithm is based on an approach due to Barvinok for approximating evaluations of a polynomial based on the location of the complex zeros and a technique due to Patel and Regts for efficiently computing the leading coefficients of graph polynomials on bounded degree graphs. Finally, we show how our algorithm can be extended to approximate certain output probability amplitudes of quantum circuits.

scholarly journals Chirality transitions in frustrated S2-valued spin systems

2016 ◽  
Vol 26 (08) ◽  
pp. 1481-1529 ◽  
Author(s):  
Marco Cicalese ◽  
Matthias Ruf ◽  
Francesco Solombrino
Keyword(s):  
Finite Number ◽  
Spin System ◽  
Spin Chain ◽  
Continuum Limit ◽  
Spin Systems ◽  
Interaction Parameters ◽  
Energy Scaling ◽  
Text Model ◽  
The Cost ◽  
Chirality Transitions

We study the discrete-to-continuum limit of the helical XY [Formula: see text]-spin system on the lattice [Formula: see text]. We scale the interaction parameters in order to reduce the model to a spin chain in the vicinity of the Landau–Lifschitz point and we prove that at the same energy scaling under which the [Formula: see text]-model presents scalar chirality transitions, the cost of every vectorial chirality transition is zero. In addition we show that if the energy of the system is modified penalizing the distance of the [Formula: see text]-field from a finite number of copies of [Formula: see text], it is still possible to prove the emergence of nontrivial (possibly trace-dependent) chirality transitions.

scholarly journals A complexity classification of spin systems with an external field

2015 ◽  
Vol 112 (43) ◽  
pp. 13161-13166 ◽  
Author(s):  
Leslie Ann Goldberg ◽  
Mark Jerrum
Keyword(s):  
Ising Model ◽  
External Field ◽  
Spin System ◽  
Spin Systems ◽  
Ferromagnetic Ising Model ◽  
Complexity Classification ◽  
Antiferromagnetic Ising Model ◽  
Computational Difficulty ◽  
State Spin

We study the computational complexity of approximating the partition function of a q-state spin system with an external field. There are just three possible levels of computational difficulty, depending on the interaction strengths between adjacent spins: (i) efficiently exactly computable, (ii) equivalent to the ferromagnetic Ising model, and (iii) equivalent to the antiferromagnetic Ising model. Thus, every nontrivial q-state spin system, irrespective of the number q of spins, is computationally equivalent to one of two fundamental two-state spin systems.

scholarly journals TWO-DIMENSIONAL LATTICE GRAVITY AS A SPIN SYSTEM

1994 ◽  
Vol 05 (02) ◽  
pp. 359-361 ◽  
Author(s):  
W. BEIRL ◽  
H. MARKUM ◽  
J. RIEDLER
Keyword(s):  
Partition Function ◽  
External Field ◽  
Path Integral ◽  
Spin System ◽  
Higher Dimensions ◽  
Two Dimensional ◽  
Path Integral Formulation ◽  
Regge Calculus ◽  
Dimensional Lattice ◽  
Matter Fields

Quantum gravity is studied in the path integral formulation applying the Regge calculus. Restricting the quadratic link lengths of the originally triangular lattice the path integral can be transformed to the partition function of a spin system with higher couplings on a Kagomé lattice. Various measures acting as external field are considered. Extensions to matter fields and higher dimensions are discussed.

scholarly journals Exact Solution of a One-dimensional Spin System

1970 ◽  
Vol 23 (5) ◽  
pp. 927 ◽  
Author(s):  
RW Gibberd
Keyword(s):  
Phase Transition ◽  
Exact Solution ◽  
Free Energy ◽  
Partition Function ◽  
Thermodynamic Limit ◽  
Spin System ◽  
Spin Systems ◽  
One Dimensional ◽  
Gibb’S Free Energy ◽  
Theory Of Superconductivity

The partition function and the Gibb's free energy are calculated exactly in the thermodynamic limit, using techniques which are well known in the theory of superconductivity. This calculation illustrates explicitly the similarity between the phase transition in superconductivity and the molecular field transitions in spin systems.

scholarly journals The Complexity of Approximating the Matching Polynomial in the Complex Plane

2021 ◽  
Vol 13 (2) ◽  
pp. 1-37
Author(s):  
Ivona Bezáková ◽  
Andreas Galanis ◽  
Leslie Ann Goldberg ◽  
Daniel Štefankovič
Keyword(s):  
Real Axis ◽  
Complex Plane ◽  
Maximum Degree ◽  
Negative Real Axis ◽  
Positive Real ◽  
Bounded Degree ◽  
Matching Polynomial ◽  
Correlation Decay ◽  
Connective Constant ◽  
General Complex

We study the problem of approximating the value of the matching polynomial on graphs with edge parameter γ, where γ takes arbitrary values in the complex plane. When γ is a positive real, Jerrum and Sinclair showed that the problem admits an FPRAS on general graphs. For general complex values of γ, Patel and Regts, building on methods developed by Barvinok, showed that the problem admits an FPTAS on graphs of maximum degree Δ as long as γ is not a negative real number less than or equal to −1/(4(Δ −1)). Our first main result completes the picture for the approximability of the matching polynomial on bounded degree graphs. We show that for all Δ ≥ 3 and all real γ less than −1/(4(Δ −1)), the problem of approximating the value of the matching polynomial on graphs of maximum degree Δ with edge parameter γ is #P-hard. We then explore whether the maximum degree parameter can be replaced by the connective constant. Sinclair et al. showed that for positive real γ, it is possible to approximate the value of the matching polynomial using a correlation decay algorithm on graphs with bounded connective constant (and potentially unbounded maximum degree). We first show that this result does not extend in general in the complex plane; in particular, the problem is #P-hard on graphs with bounded connective constant for a dense set of γ values on the negative real axis. Nevertheless, we show that the result does extend for any complex value γ that does not lie on the negative real axis. Our analysis accounts for complex values of γ using geodesic distances in the complex plane in the metric defined by an appropriate density function.

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