**1. Introduction**

In 1985, Webb et al. [1] presented their pioneering experiment, showing Aharonov-Bohm oscillations in a nanoscopic metallic ring and a role of quantum interference (QI) in electron transport. Later, Ji et al. [2] demonstrated the electronic analogue of the optical Mach–Zehnder interferometer (MZI), which was based on closed-geometry transport through single edge states in the quantum Hall regime. Theoretical studies [3–7] predicted coherent transport through single molecules with a ring structure, where, due to their small size, one could show constructive or destructive quantum interference effects at room temperatures. From 2011, these predictions have been experimentally verified, using mechanically controllable break junction (MCBJ) and scanning tunneling microscope break junction (STM-BJ) techniques [8,9] in various molecular systems: Single phenyl, polycyclic aromatic, and conjugated heterocyclic blocks, as well as hydrocarbons (for a recent review on QI in molecular junctions, see [10,11] and the references therein).

Our interest is in the internal local currents and their correlations in a ring geometry to see a role of quantum interference. An interesting aspect is the formation of a quantum vortex flow driven by a net current from the source to the drain electrode, which has been studied in many molecular systems [7,10,12–21] (see also [22]). It has also been shown that, under some conditions, a circular thermoelectric current can exceed the transport current [23]. In particular, our studies focus on the role of the states with opposite chirality in the ring and on the QI effect and the circular current. Correlations of the electron currents (shot noise) through edge states in the Mach–Zehnder interferometer have been extensively studied by Buttiker et al. [24–28] (see also [29] and the references therein). However, in a metallic (or molecular) ring, the situation is different than in the MZI, as multiple reflections are relevant to the formation of the circular current. Our studies will show that the transition from laminar to vortex flow is manifested in the shot noise of local currents. In particular, it will be seen in a cross-correlation function for the currents in different branches of the ring, which becomes negative and large in the presence of the circular current.

The paper is organized as follows. In the next chapter, Section 2, we will present the model of three quantum dots in a ring geometry, which is the simplest model showing all aspects of QI and current correlations. The model includes a magnetic flux threading the ring, which changes interference conditions as well as inducing a persistent current. The net transport current and the local bond currents, as well as the persistent current (and their conductances), are derived analytically, by means of the non-equilibrium Keldysh Green function technique. It will be shown that the correlation function for the net transport current can be expressed as a composition of the correlation functions for the local currents inside the ring. We will, also, show all shot noise components; in particular, the one for the net transport current (given by Lesovik's formula [30]). The next chapters, Sections 3–5, present the analyses of the results for the case Φ = 0 (without the magnetic flux), for the case with the persistent current only (without the source-drain bias *V*), and for the general case (for *V* = 0, Φ = 0) showing the interplay between the bond currents and the persistent current. Finally, in Section 6, the main results of the paper are summarized.

### **2. Calculations of Currents and Their Correlations in Triangular Quantum Dot System**
