## The Annals of Probability

### An $L^{p}$ theory of sparse graph convergence II: LD convergence, quotients and right convergence

#### Abstract

We extend the $L^{p}$ theory of sparse graph limits, which was introduced in a companion paper, by analyzing different notions of convergence. Under suitable restrictions on node weights, we prove the equivalence of metric convergence, quotient convergence, microcanonical ground state energy convergence, microcanonical free energy convergence and large deviation convergence. Our theorems extend the broad applicability of dense graph convergence to all sparse graphs with unbounded average degree, while the proofs require new techniques based on uniform upper regularity. Examples to which our theory applies include stochastic block models, power law graphs and sparse versions of $W$-random graphs.

#### Article information

Source
Ann. Probab., Volume 46, Number 1 (2018), 337-396.

Dates
Revised: March 2017
First available in Project Euclid: 5 February 2018

https://projecteuclid.org/euclid.aop/1517821225

Digital Object Identifier
doi:10.1214/17-AOP1187

Mathematical Reviews number (MathSciNet)
MR3758733

Zentralblatt MATH identifier
06865125

Subjects
Secondary: 82B99: None of the above, but in this section

#### Citation

Borgs, Christian; Chayes, Jennifer T.; Cohn, Henry; Zhao, Yufei. An $L^{p}$ theory of sparse graph convergence II: LD convergence, quotients and right convergence. Ann. Probab. 46 (2018), no. 1, 337--396. doi:10.1214/17-AOP1187. https://projecteuclid.org/euclid.aop/1517821225

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