scholarly journals Counting Homomorphisms to Trees Modulo a Prime

2021 ◽  
Vol 13 (3) ◽  
pp. 1-33
Author(s):  
Andreas Göbel ◽  
J. A. Gregor Lagodzinski ◽  
Karen Seidel
Keyword(s):  
Statistical Physics ◽  
Independent Sets ◽  
Partition Functions ◽  
Interim Result ◽  
Graph Homomorphism ◽  
Graph Theoretic ◽  
Modular Counting ◽  
Graph Homomorphisms ◽  
Counting Functions ◽  
The Common

Many important graph-theoretic notions can be encoded as counting graph homomorphism problems, such as partition functions in statistical physics, in particular independent sets and colourings. In this article, we study the complexity of  # p H OMS T O H , the problem of counting graph homomorphisms from an input graph to a graph H modulo a prime number  p . Dyer and Greenhill proved a dichotomy stating that the tractability of non-modular counting graph homomorphisms depends on the structure of the target graph. Many intractable cases in non-modular counting become tractable in modular counting due to the common phenomenon of cancellation. In subsequent studies on counting modulo 2, however, the influence of the structure of  H on the tractability was shown to persist, which yields similar dichotomies. Our main result states that for every tree  H and every prime  p the problem # p H OMS T O H is either polynomial time computable or # p P-complete. This relates to the conjecture of Faben and Jerrum stating that this dichotomy holds for every graph H when counting modulo 2. In contrast to previous results on modular counting, the tractable cases of # p H OMS T O H are essentially the same for all values of the modulo when H is a tree. To prove this result, we study the structural properties of a homomorphism. As an important interim result, our study yields a dichotomy for the problem of counting weighted independent sets in a bipartite graph modulo some prime  p . These results are the first suggesting that such dichotomies hold not only for the modulo 2 case but also for the modular counting functions of all primes  p .

scholarly journals Automatic contraction of unstructured tensor networks

2020 ◽  
Vol 8 (1) ◽  
Author(s):  
Adam Jermyn
Keyword(s):  
Partition Function ◽  
Statistical Physics ◽  
Discrete Models ◽  
Correlation Structure ◽  
Central Problem ◽  
Partition Functions ◽  
Lattice Systems ◽  
Tensor Networks ◽  
Tensor Network

The evaluation of partition functions is a central problem in statistical physics. For lattice systems and other discrete models the partition function may be expressed as the contraction of a tensor network. Unfortunately computing such contractions is difficult, and many methods to make this tractable require periodic or otherwise structured networks. Here I present a new algorithm for contracting unstructured tensor networks. This method makes no assumptions about the structure of the network and performs well in both structured and unstructured cases so long as the correlation structure is local.

scholarly journals Quantum mechanism of condensation and high Tc superconductivity

2019 ◽  
Vol 33 (14) ◽  
pp. 1950139
Author(s):  
Tian Ma ◽  
Shouhong Wang
Keyword(s):  
Statistical Physics ◽  
Quantum Physics ◽  
Average Energy ◽  
Physics First ◽  
Electron Pairs ◽  
High Tc ◽  
Quantum Mechanism ◽  
Recent Developments ◽  
The Common ◽  
New Interpretation

The main objective of this paper is to introduce a new quantum mechanism of condensates and superconductivity based on a new interpretation of quantum mechanical wavefunctions, and on recent developments in quantum physics and statistical physics. First, we postulate that the wavefunction [Formula: see text]e[Formula: see text] is the common wavefunction for all particles in the same class determined by the external potential V(x), [Formula: see text](x)[Formula: see text] represents the distribution density of the particles, and [Formula: see text] is the velocity field of the particles. Although the new interpretation does not alter the basic theories of quantum mechanics, it is an entirely different interpretation from the classical Bohr interpretation, removes all absurdities and offers new insights for quantum physics and for condensed matter physics. Second, we show that the key for condensation of bosonic particles is that their interaction is sufficiently weak to ensure that a large collection of boson particles are in a state governed by the same condensation wavefunction field [Formula: see text] under the same bounding potential V. For superconductivity, the formation of superconductivity comes down to conditions for the formation of electron pairs, and for the electron pairs to share a common wavefunction. Thanks to the recently developed principle of interaction dynamics (PID) interaction potential of electrons and the average-energy level formula of temperature, these conditions for superconductivity are explicitly derived. Furthermore, we obtain both microscopic and macroscopic formulas for the critical temperature. The field and topological phase transition equations for condensates are also derived.

scholarly journals Solving Graph Homomorphism and Subgraph Isomorphism Problems Faster Through Clique Neighbourhood Constraints

2021 ◽  
Author(s):  
Sonja Kraiczy ◽  
Ciaran McCreesh
Keyword(s):  
Constraint Programming ◽  
State Of The Art ◽  
Programming Approach ◽  
Subgraph Isomorphism ◽  
Graph Homomorphism ◽  
General Graph ◽  
Graph Homomorphisms ◽  
Practical Constraint ◽  
New Form ◽  
Programming Algorithms

Graph homomorphism problems involve finding adjacency-preserving mappings between two given graphs. Although theoretically hard, these problems can often be solved in practice using constraint programming algorithms. We show how techniques from the state-of-the-art in subgraph isomorphism solving can be applied to broader graph homomorphism problems, and introduce a new form of filtering based upon clique-finding. We demonstrate empirically that this filtering is effective for the locally injective graph homomorphism and subgraph isomorphism problems, and gives the first practical constraint programming approach to finding general graph homomorphisms.

scholarly journals Quantum speedup of Monte Carlo methods

2015 ◽  
Vol 471 (2181) ◽  
pp. 20150301 ◽  
Author(s):  
Ashley Montanaro
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Methods ◽  
Statistical Physics ◽  
Probability Distributions ◽  
General Setting ◽  
Quantum Algorithm ◽  
Quantum Walks ◽  
Partition Functions ◽  
Performance Bounds ◽  
Quantum Speedup

Monte Carlo methods use random sampling to estimate numerical quantities which are hard to compute deterministically. One important example is the use in statistical physics of rapidly mixing Markov chains to approximately compute partition functions. In this work, we describe a quantum algorithm which can accelerate Monte Carlo methods in a very general setting. The algorithm estimates the expected output value of an arbitrary randomized or quantum subroutine with bounded variance, achieving a near-quadratic speedup over the best possible classical algorithm. Combining the algorithm with the use of quantum walks gives a quantum speedup of the fastest known classical algorithms with rigorous performance bounds for computing partition functions, which use multiple-stage Markov chain Monte Carlo techniques. The quantum algorithm can also be used to estimate the total variation distance between probability distributions efficiently.

scholarly journals Geometry, Coordinatization and Cardinality of the Rational Numbers from Physical Perspective

2021 ◽  
Vol 2090 (1) ◽  
pp. 012037
Author(s):  
Kaushik Ghosh
Keyword(s):  
Line Segment ◽  
Statistical Physics ◽  
Rational Numbers ◽  
Partition Functions ◽  
Actual Process ◽  
Real Numbers ◽  
Hausdorff Topology ◽  
Straight Line ◽  
Definition Of ◽  
Positive Integers

Abstract In this article, we will first discuss the completeness of real numbers in the context of an alternate definition of the straight line as a geometric continuum. According to this definition, points are not regarded as the basic constituents of a line segment and a line segment is considered to be a fundamental geometric object. This definition is in particular suitable to coordinatize different points on the straight line preserving the order properties of real numbers. Geometrically fundamental nature of line segments are required in physical theories like the string theory. We will construct a new topology suitable for this alternate definition of the straight line as a geometric continuum. We will discuss the cardinality of rational numbers in the later half of the article. We will first discuss what we do in an actual process of counting and define functions well-defined on the set of all positive integers. We will follow an alternate approach that depends on the Hausdorff topology of real numbers to demonstrate that the set of positive rationals can have a greater cardinality than the set of positive integers. This approach is more consistent with an actual act of counting. We will illustrate this aspect further using well-behaved functionals of convergent functions defined on the finite dimensional Cartezian products of the set of positive integers and non-negative integers. These are similar to the partition functions in statistical physics. This article indicates that the axiom of choice can be a better technique to prove theorems that use second-countability. This is important for the metrization theorems and physics of spacetime.

scholarly journals Information Flows, Graphs and their Guessing Numbers

10.37236/962 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Søren Riis
Keyword(s):  
Network Coding ◽  
Circuit Complexity ◽  
Information Flows ◽  
Boolean Circuit ◽  
Graph Theoretic ◽  
Common Information ◽  
Shift Problem ◽  
Cyclic Shifts ◽  
The Common ◽  
Multiuser Information Theory

Valiant's shift problem asks whether all $n$ cyclic shifts on $n$ bits can be realized if $n^{(1+\epsilon)}$ input output pairs ($\epsilon < 1$) are directly connected and there are additionally $m$ common bits available that can be arbitrary functions of all the inputs. If it could be shown that this is not realizable with $m = O({n\over \log \log n})$ common bits then a significant breakthrough in Boolean circuit complexity would follow. In this paper it is shown that in certain cases all cyclic shifts are realizable with $m=(n- n^{\epsilon})/2$ common bits. Previously, no solution with $m < n - o(n)$ was known, and Valiant had conjectured that $m < n/2$ was not achievable. The construction therefore establishes a novel way of realizing communication in the manner of Network Coding for the shift problem, but leaves the viability of the common information approach to proving lower bounds in Circuit Complexity open. The construction uses the graph-theoretic notion of guessing number. As a by-product the paper also establish an interesting link between Circuit Complexity and Network Coding, a new direction of research in multiuser information theory.

scholarly journals COMMON SOURCE OF NUMEROUS THETA FUNCTION IDENTITIES

2007 ◽  
Vol 49 (1) ◽  
pp. 61-79 ◽  
Author(s):  
CHU WENCHANG
Keyword(s):  
Difference Equation ◽  
Recent Work ◽  
Theta Function ◽  
Infinite Order ◽  
Partition Functions ◽  
Common Source ◽  
The Common ◽  
Theta Function Identities

Abstract.Motivated by the recent work due to Warnaar (2005), two new and elementary proofs are presented for a very useful q-difference equation on eight shifted factorials of infinite order. As the common source of theta function identities, this q-difference equation is systematically explored to review old and establish new identities on Ramanujan's partition functions. Most of the identities obtained can be interpreted in terms of theorems on classical partitions.

scholarly journals Perfect matchings, rank of connection tensors and graph homomorphisms

2021 ◽  
pp. 1-36
Author(s):  
Jin-Yi Cai ◽  
Artem Govorov
Keyword(s):  
General Setting ◽  
Perfect Matchings ◽  
Edge Weight ◽  
Graph Algebras ◽  
Graph Homomorphism ◽  
Graph Homomorphisms ◽  
Graph Properties ◽  
Connection Matrices ◽  
Holant Problems ◽  
Necessary And Sufficient

Abstract We develop a theory of graph algebras over general fields. This is modelled after the theory developed by Freedman et al. (2007, J. Amer. Math. Soc.20 37–51) for connection matrices, in the study of graph homomorphism functions over real edge weight and positive vertex weight. We introduce connection tensors for graph properties. This notion naturally generalizes the concept of connection matrices. It is shown that counting perfect matchings, and a host of other graph properties naturally defined as Holant problems (edge models), cannot be expressed by graph homomorphism functions with both complex vertex and edge weights (or even from more general fields). Our necessary and sufficient condition in terms of connection tensors is a simple exponential rank bound. It shows that positive semidefiniteness is not needed in the more general setting.

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.

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