The complexity of approximating the complex-valued Ising model on bounded degree graphs

by   Andreas Galanis, et al.

We study the complexity of approximating the partition function Z_Ising(G; β) of the Ising model in terms of the relation between the edge interaction β and a parameter Δ which is an upper bound on the maximum degree of the input graph G. Following recent trends in both statistical physics and algorithmic research, we allow the edge interaction β to be any complex number. Many recent partition function results focus on complex parameters, both because of physical relevance and because of the key role of the complex case in delineating the tractability/intractability phase transition of the approximation problem. In this work we establish both new tractability results and new intractability results. Our tractability results show that Z_Ising(-; β) has an FPTAS when |β - 1 | / |β + 1 | < tan(π / (4 Δ - 4)). The core of the proof is showing that there are no inputs G that make the partition function 0 when β is in this range. Our result significantly extends the known zero-free region of the Ising model (and hence the known approximation results). Our intractability results show that it is #P-hard to multiplicatively approximate the norm and to additively approximate the argument of Z_Ising(-; β) when β∈ℂ is an algebraic number such that β∉ℝ∪{i, -i} and |β - 1| / |β + 1 | > 1 / √(Δ - 1). These are the first results to show intractability of approximating Z_Ising(-, β) on bounded degree graphs with complex β. Moreover, we demonstrate situations in which zeros of the partition function imply hardness of approximation in the Ising model.



page 1

page 2

page 3

page 4


Lee-Yang zeros and the complexity of the ferromagnetic Ising Model on bounded-degree graphs

We study the computational complexity of approximating the partition fun...

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

We study the problem of approximating the Ising model partition function...

The complexity of approximating the complex-valued Potts model

We study the complexity of approximating the partition function of the q...

A note on the high-fugacity hard-core model on bounded-degree bipartite expander graphs

Jenssen, Keevash and Perkins give an FPTAS and an efficient sampling alg...

On zero-free regions for the anti-ferromagnetic Potts model on bounded-degree graphs

For a graph G=(V,E), k∈N, and a complex number w the partition function ...

Location of zeros for the partition function of the Ising model on bounded degree graphs

The seminal Lee-Yang theorem states that for any graph the zeros of the ...

On the Partition Function and Random Maximum A-Posteriori Perturbations

In this paper we relate the partition function to the max-statistics of ...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.