A reverse Gaussian correlation inequality by adding cones

2017 ◽  
Vol 123 ◽  
pp. 84-87
Author(s):  
Xiaohong Chen ◽  
Fuchang Gao
Keyword(s):  
Correlation Inequality

scholarly journals A useful elementary correlation inequality

1989 ◽  
Vol 50 (2) ◽  
pp. 305-307 ◽  
Author(s):  
Ravi Boppona ◽  
Joel Spencer
Keyword(s):  
Correlation Inequality

scholarly journals Sidorenko's Conjecture, Colorings and Independent Sets

10.37236/6019 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Péter Csikvári ◽  
Zhicong Lin
Keyword(s):  
Bipartite Graph ◽  
Complete Graph ◽  
Independent Sets ◽  
Correlation Inequality

Let $\hom(H,G)$ denote the number of homomorphisms from a graph $H$ to a graph $G$. Sidorenko's conjecture asserts that for any bipartite graph $H$, and a graph $G$ we have$$\hom(H,G)\geq v(G)^{v(H)}\left(\frac{\hom(K_2,G)}{v(G)^2}\right)^{e(H)},$$where $v(H),v(G)$ and $e(H),e(G)$ denote the number of vertices and edges of the graph $H$ and $G$, respectively. In this paper we prove Sidorenko's conjecture for certain special graphs $G$: for the complete graph $K_q$ on $q$ vertices, for a $K_2$ with a loop added at one of the end vertices, and for a path on $3$ vertices with a loop added at each vertex. These cases correspond to counting colorings, independent sets and Widom-Rowlinson colorings of a graph $H$. For instance, for a bipartite graph $H$ the number of $q$-colorings $\mathrm{ch}(H,q)$ satisfies$$\mathrm{ch}(H,q)\geq q^{v(H)}\left(\frac{q-1}{q}\right)^{e(H)}.$$In fact, we will prove that in the last two cases (independent sets and Widom-Rowlinson colorings) the graph $H$ does not need to be bipartite. In all cases, we first prove a certain correlation inequality which implies Sidorenko's conjecture in a stronger form.

scholarly journals A correlation inequality for connection events in percolation

2001 ◽  
Vol 29 (1) ◽  
pp. 123-126 ◽  
Author(s):  
J. van den Berg ◽  
J. Kahn
Keyword(s):  
Correlation Inequality

A correlation inequality for bipartite graphs

1993 ◽  
Vol 9 (2-4) ◽  
pp. 201-204 ◽  
Author(s):  
Alexander Sidorenko
Keyword(s):  
Bipartite Graphs ◽  
Correlation Inequality

Correlation and variability in birth processes

1993 ◽  
Vol 30 (02) ◽  
pp. 275-284
Author(s):  
Peter Donnelly ◽  
Thomas Kurtz ◽  
Paul Marjoram
Keyword(s):  
Negative Correlation ◽  
Correlation Structure ◽  
Linear Case ◽  
Birth Process ◽  
Correlation Inequality ◽  
Birth Rates ◽  
Birth Processes

Faddy (1990) has conjectured that the variability of a pure birth process is increased, relative to the linear case, if the birth rates are convex and decreased if they are concave. We prove the conjecture by relating variability to the correlation structure of certain more informative versions of the process. A correlation inequality due to Harris (1977) is used to derive the necessary positive and negative correlation results.

On a Correlation Inequality of Farr

1992 ◽  
Vol 1 (2) ◽  
pp. 157-160 ◽  
Author(s):  
Colin McDiarmid
Keyword(s):  
Simple Proof ◽  
Random Set ◽  
Correlation Inequality

Suppose that each vertex of a graph independently chooses a colour uniformly from the set {1, …, k}; and let Si be the random set of vertices coloured i. Farr shows that the probability that each set Si is stable (so that the colouring is proper) is at most the product of the k probabilities that the sets Si separately are stable. We give here a simple proof of an extension of this result.

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