This is a commonly misunderstood concept in amplifiers. What is meant by 'high current'?
Does it mean that the amplifier behaves as a voltage source, and so can double power as the speaker load is cut in half or does it mean something else?
It means both but often not at the same time.
However it might be helpful to run some math once you see the advertised 'current' figures for a particular amplifier. I remember one amplifier advertised 80 amps. That seemed a bit crazy to me, so I ran the math, and gave the amp the benefit of the doubt. Here is what I came up with:
The Power formula is Power=Current squared times Resistance.
So if we have 80 amps and a one ohm load, the resulting power is 6400 watts (the 'benefit of the doubt was that I allowed it to drive one ohm- do the math for four ohms and you see what I mean). The amp in question did not claim to make any such power, so its obvious that the current rating meant something else.
What it means is the amount of current that is seen when the power supply is shorted out for 10 milliseconds. It is a measure of power supply capacity and nothing more. Can this affect authority in the amp? Sure- as the additional capacity can reduce intermodulation distortion at higher power levels. That has a direct effect on impact.
Just to give an additional bit of information- to make 1000 watts into 4 ohms the amplifier will need to make 15.81 amps. That's all!
IOW, most of the time when you see that expression- 'high current' it has nothing to do with the amp's ability to deliver power. And ultimately it is power that drives the speaker, and power is the combination of current *and* voltage.
You also have to be careful of the speaker. Not all speakers like an amplifier with a super low output impedance- some will play bass with more extension and impact if the amp has a higher output impedance (and such an amplifier might not be described as 'high current'). This is because there is more to it that just simple low output impedance in the amp; there is some physics involved.
For more about this see
http://www.atma-sphere.com/Resources/Paradigms_in_Amplifier_Design.php