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The figure below is taken from Reath, K. et al. (2019)
A couple of things can be seen from this representation
Volcanism seems to be strongly correlated with subduction zones, in this case the Nazca plate and South America plates.
There is a significant gap of volcanoes in both Peru and Chile. Which suggests that subduction is not the whole story. There must be something else happening
Question is: do we know what? What is the reason for these gaps?
As you said, the Andean belt is divided into four segments, usually called the northern, central, southern, and austral volcanic zones (NVZ, CVZ, SVZ and AVZ, respectively; your map lacks the AVZ). This has been interpreted as a difference in the angle of subduction. Under the active zones, the subducting plate (called "slab") sinks into the mantle, releasing water, and the hydrated mantle partially melts, a process called "metasomatism". But under the gaps, the slab has a low angle ("flat slab subduction"), so it doesn't sink deep enough to trigger the melting process. There is no partial melting of the mantle under these zones.
Now, why has the slab failed to sink under these zones? This is due to the presence of submarine ridges. Between the NVZ and CVZ, there is the Nazca Ridge. Similarly, there is the Juan Fernandez ridge between the CVZ and SVZ. Those ridges are young, making the oceanic crust less dense, thus more buoyant (usually it is old, dense oceanic crust that subducts). The ridges also make the oceanic crust thicker, thus harder to subduct.
Image source; by Chiton magnificus in 2011 and placed in the public domain.
Two papers if you want to go further:
So, this is the setting. A relatively young plate plate is subducted under the south American continent. The subducting oceanic plate contains water that is squeezed out under pressure and starts to rise. Water (fluids) generally lower the melting point, so hot asthenosphere above the colder subducting plate partially melts, the magma, being more mobile, begins to rise and eventually reaches the surface, forming a volcanic arc that sketches the subduction zone on the suurface.
So why are there volcanoes in some parts of the subducting plate, and not in others ? This is a temporary state. At present ridges are subducted in these zones. These ridges are warm and less buoyant than the surrounding plate, they don't pass the asthenosphere but slide (after a short dip) (near) horizontally under the continental crust, causing uplift and orogeny (mountain rise). In the past, there has been volcanism in these zones as well, when steeper subduction took place.
So, yes, subduction is the whole story, but the type of subduction plays a role. The last link describes the current understanding of how these areas with volcanism will shift with subduction of the ridges, as well as sketches of the subduction types, as pictured by seismics.
When an oceanic plate is subducted below a continental plate, it dives into the mantle, taking water with it and producing frictional heat which melts some of the rock (rock with a high water content melts more easily). This produces plumes of magma beneath the continental plate margins. As the magma is lighter than the mantle material, it rises until it meets the continental plate above. The plumes try to break through, but are not always successful. Those that break through form volcanoes, those that don't quite manage to break through form plutons.
Looking at your map, we see gaps where the magma plumes have so far failed to break through, perhaps in places where the crust was thicker. In places where it seems there should be more volcanoes, there could be plutons marking a failed attempt, or there could still be a rising plume which might one day succeed in reaching the surface. Close to a line of volcanoes, it could be that these volcanoes are bleeding off so much of the rising magma that the remainder finds it hard to break through to the surface.
