scholarly journals EIGENVALUES AND LINEAR QUASIRANDOM HYPERGRAPHS

2015 ◽  
Vol 3 ◽  
Author(s):  
JOHN LENZ ◽  
DHRUV MUBAYI
Keyword(s):  
Partition Function ◽  
Spectral Gap ◽  
Spectral Approach ◽  
Uniform Hypergraph ◽  
Largest Eigenvalues ◽  
Linear Hypergraph ◽  
Partial Extension

Let$p(k)$denote the partition function of$k$. For each$k\geqslant 2$, we describe a list of$p(k)-1$quasirandom properties that a$k$-uniform hypergraph can have. Our work connects previous notions on linear hypergraph quasirandomness by Kohayakawa, Rödl, and Skokan, and by Conlon, Hàn, Person, and Schacht, and the spectral approach of Friedman and Wigderson. For each of the quasirandom properties that is described, we define the largest and the second largest eigenvalues. We show that a hypergraph satisfies these quasirandom properties if and only if it has a large spectral gap. This answers a question of Conlon, Hàn, Person, and Schacht. Our work can be viewed as a partial extension to hypergraphs of the seminal spectral results of Chung, Graham, and Wilson for graphs.

scholarly journals Free partition functions and an averaged holographic duality

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Nima Afkhami-Jeddi ◽  
Henry Cohn ◽  
Thomas Hartman ◽  
Amirhossein Tajdini
Keyword(s):  
Partition Function ◽  
Spectral Gap ◽  
Central Charge ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Partition Functions ◽  
General Constraints ◽  
Dimensional Gravity ◽  
Holographic Duality

Abstract We study the torus partition functions of free bosonic CFTs in two dimensions. Integrating over Narain moduli defines an ensemble-averaged free CFT. We calculate the averaged partition function and show that it can be reinterpreted as a sum over topologies in three dimensions. This result leads us to conjecture that an averaged free CFT in two dimensions is holographically dual to an exotic theory of three-dimensional gravity with U(1)c×U(1)c symmetry and a composite boundary graviton. Additionally, for small central charge c, we obtain general constraints on the spectral gap of free CFTs using the spinning modular bootstrap, construct examples of Narain compactifications with a large gap, and find an analytic bootstrap functional corresponding to a single self-dual boson.

scholarly journals Some Applications of Wagner's Weighted Subgraph Counting Polynomial

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Ferenc Bencs ◽  
Péter Csikvári ◽  
Guus Regts
Keyword(s):  
Ising Model ◽  
Partition Function ◽  
Line Graphs ◽  
Uniform Hypergraph ◽  
Edge Cover ◽  
Absolute Value ◽  
On Line ◽  
Ferromagnetic Ising Model ◽  
Cover Polynomial ◽  
Counting Polynomial

We use Wagner's weighted subgraph counting polynomial to prove that the partition function of the anti-ferromagnetic Ising model on line graphs is real rooted and to prove that roots of the edge cover polynomial have absolute value at most $4$. We more generally show that roots of the edge cover polynomial of a $k$-uniform hypergraph have absolute value at most $2^k$, and discuss applications of this to the roots of domination polynomials of graphs. We moreover discuss how our results relate to efficient algorithms for approximately computing evaluations of these polynomials.  

scholarly journals The first two largest eigenvalues of Laplacian, spectral gap problem and Cheeger constant of graphs

2017 ◽  
Author(s):  
Opiyo Samuel ◽  
Yudi Soeharyadi ◽  
Marcus Wono Setyabudhi
Keyword(s):  
Spectral Gap ◽  
Cheeger Constant ◽  
Largest Eigenvalues

scholarly journals On $r$-Uniform Linear Hypergraphs with no Berge-$K_{2,t}$

10.37236/6470 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Craig Timmons
Keyword(s):  
Asymptotic Formula ◽  
Uniform Hypergraph ◽  
Linear Hypergraphs ◽  
Linear Hypergraph

Let $\mathcal{F}$ be an $r$-uniform hypergraph and $G$ be a multigraph. The hypergraph $\mathcal{F}$ is a Berge-$G$ if there is a bijection $f: E(G) \rightarrow E( \mathcal{F} )$ such that $e \subseteq f(e)$ for each $e \in E(G)$.  Given a family of multigraphs $\mathcal{G}$, a hypergraph $\mathcal{H}$ is said to be $\mathcal{G}$-free if for each $G \in \mathcal{G}$, $\mathcal{H}$ does not contain a subhypergraph that is isomorphic to a Berge-$G$. We prove bounds on the maximum number of edges in an $r$-uniform linear hypergraph that is $K_{2,t}$-free. We also determine an asymptotic formula for the maximum number of edges in a linear 3-uniform 3-partite hypergraph that is $\{C_3 , K_{2,3} \}$-free. 

scholarly journals Estimating Gibbs partition function with quantum Clifford sampling

2022 ◽  
Author(s):  
Yusen Wu ◽  
Jingbo B Wang
Keyword(s):  
Partition Function ◽  
Spectral Gap ◽  
Sampling Technique ◽  
Quantum Circuit ◽  
Quantum Systems ◽  
Quantum Circuits ◽  
Partition Functions ◽  
Stochastic Matrices ◽  
Many Body ◽  
Quantum Devices

Abstract The partition function is an essential quantity in statistical mechanics, and its accurate computation is a key component of any statistical analysis of quantum system and phenomenon. However, for interacting many-body quantum systems, its calculation generally involves summing over an exponential number of terms and can thus quickly grow to be intractable. Accurately and efficiently estimating the partition function of its corresponding system Hamiltonian then becomes the key in solving quantum many-body problems. In this paper we develop a hybrid quantumclassical algorithm to estimate the partition function, utilising a novel Clifford sampling technique. Note that previous works on quantum estimation of partition functions require O(1/ε√∆)-depth quantum circuits [17, 23], where ∆ is the minimum spectral gap of stochastic matrices and ε is the multiplicative error. Our algorithm requires only a shallow O(1)-depth quantum circuit, repeated O(n/ε2) times, to provide a comparable ε approximation. Shallow-depth quantum circuits are considered vitally important for currently available NISQ (Noisy Intermediate-Scale Quantum) devices.

From Partition Function to Phase Diagram — Statistical Thermodynamics of the LaNi5—H System*

1992 ◽  
Vol 1 (1) ◽  
pp. 47-57
Author(s):  
H. Brodowsky ◽  
K. Yasuda ◽  
K. Itagaki
Keyword(s):  
Phase Diagram ◽  
Partition Function ◽  
Statistical Thermodynamics

CALCULATION OF ISOTOPIC REDUCED PARTITION FUNCTION RATIO FOR AQUA COMPLEXES OF MAGNESIUM CATION

2017 ◽  
pp. 138-145
Author(s):  
A.V. BOCHKAREV ◽  
◽  
S.L. BELOPUKHOV ◽  
A.V. ZHEVNEROV ◽  
S.V. DEMIN ◽  
...  
Keyword(s):  
Partition Function ◽  
Aqua Complexes ◽  
Magnesium Cation

A canonical ensemble derivation of the McMillan-Mayer solution theory

1983 ◽  
Vol 48 (10) ◽  
pp. 2888-2892 ◽  
Author(s):  
Vilém Kodýtek
Keyword(s):  
Free Energy ◽  
Partition Function ◽  
Energy Function ◽  
Special Function ◽  
Helmholtz Free Energy ◽  
Canonical Ensemble ◽  
Free Energy Function ◽  
Solution Theory ◽  
Pure Solvent ◽  
The Common

A special free energy function is defined for a solution in the osmotic equilibrium with pure solvent. The partition function of the solution is derived at the McMillan-Mayer level and it is related to this special function in the same manner as the common partition function of the system to its Helmholtz free energy.

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