General Concepts > s.a. Covariance Matrix.
* Correlation functions: Given a stochastic process described by variables xi, the correlation function between two variables xi and xj is

C(xi, xj):= $$\langle x_i\, x_j\rangle - \langle x_i\rangle \langle x_j\rangle$$,

which vanishes if the variables are statistically uncorrelated.
> Related topics: see number theory [continued-fraction representation].

Correlations in Physics > s.a. Correlation Length; molecular physics [branched polymers]; stochastic processes [auto-correlation functions].
* In field theory: The field-field 2-point correlation function is often identified with the Green function for the theory; > s.a. N-point functions.
* Origin of correlations: In all non-quantum contexts, correlations arise from one of two mechanism, either a first event influences a second one by sending information (encoded in bosons or molecules or other physical carriers, depending on the particular context), or the correlated events have some common causes in their common past.
@ General references: Rajagopal & Rendell PRA(05)qp [density-matrix formulation]; Torquato IECR-cm/06 [realizable, random media].
@ Multipartite correlations: Grudka et al a0802 [classical]; Abbott et al PRA(16)-a1608 [polytopes and inequalities].
@ Decay of correlations: Lavenda JMP(82); Xu et al CSF(04) [maps and chaos]; Kastoryano & Eisert JMP(13)-a1303 [and mixing]; > s.a. entanglement entropy.
@ Related topics: Lenard CMP(73) [statistical state determined by correlations]; Dubessy NJP(14)-a1410 [statistical correlations used to image collective excitatons of an ultracold gas].
@ Specific types of systems: Pozas-Kerstjens et al a1904 [networks]; > s.a. cosmological perturbations; galaxy distribution; spin models.

In Quantum Systems > see entangled systems; quantum correlations and types of quantum correlations; quantum systems [3-qubit systems].

Quantum vs Classical Correlations > s.a. quantum discord.
* Rem: Quantum correlations appear only in the presence of classical ones, while the converse is not always true.
@ General references: Hepp CMP(74) [classical limit]; Peres AJP(78)jul [and Bell inequalities]; Henderson & Vedral JPA(01)qp; Mermin qp/02; Beltrametti & Bugajski IJTP(04); Cabello PRA(05)qp/04 [and bounds]; Svozil UJP-qp/05 [quantum correlations exceeding classical expectations]; Audenaert & Plenio NJP(06)qp; Modi & Gu IJMPB(12)-a0902 [coherent and incoherent]; Guo et al JPA(12); Ghirardi & Romano IJMPB(13)-a1205 [classical, quantum and superquantum correlations]; Wu et al SRep(14)-a1301; Walczak et al a1303 [quantum correlation dominance].
@ And causality: Oreshkov et al nComm(12)-a1105; Branciard et al NJP(16)-a1508 [causal inequalities on correlations]; Ibnouhsein PhD(14)-a1510; Miklin et al NJP(17)-a1706 [entropic approach]; Salazar et al a1712; Jia a1806 [indefinite causal structures and reduction of correlations]; Chaudhuri & Loganayagam a1807 [out-of-time-order correlations]; > s.a. bounds on quantum correlations.
@ Unified framework: Acín et al PRL(10)-a0911; Modi et al PRL(10)-a0911; Modi & Vedral AIP(11)-a1104; Geller & Piani JPA(14)-a1401.
@ Non-classical: Bellomo et al PRA(08)-a0806 [non-classically-reproducible]; Marek et al PRA(09) [in phase space]; Piani et al PRL(11)-a1106 [and distillable entanglement]; Piani & Adesso PRA(12)-a1110 [quantumness and entanglement]; Rosset et al NJP(13)-a1211 [no-go result for classical simulation].
@ Other measures of quantumness: Usha Devi & Rajagopal PRL(08)-a0707; Pankowski & Synak-Radtke JPA(08); SaiToh et al PRA(08), IJQI(08)-a0802-proc, QIC(11)-a0802; Wu et al PRA(09) [and complementarity]; Rossignoli et al PRA(10) [entropic measures]; Modi et al RMP(12)-a1112 [and the classical-quantum boundary, rev]; Auccaise et al PRL(11) [experimental witness]; Gharibian PRA(12)-a1202, in a1301-PhD [based on local unitary operations]; Farace et al NJP(14)-a1402; Rigovacca et al PRA(15)-a1507, Carmeli et al PRL(16)-a1510 [for bipartite states]; Debarba et al PRA(17)-a1611 [for fermionic systems]; Muthuganesan & Chandrasekar QIP(19)-a1904 [affinity].