Welcome to the board,.
The short answer is yes, they do breathe better, While Jim is correct, position on the body is important for any DH reg, a properly positioned DA or RAM in it's original condition can come no where near the performance of a AK. There are a number of reasons for this. First off the DA is an unbalanced first stage so it's not really in the same league. Unless the RAM you were diving has been rebuild with the current generation of reproduction parts it's way behind the curve to start with as well. The reproduction silicone main diaphragm is a vast improvement over the original design (the Kraken uses the latest version), it alone vastly improves the performance of a DA or RAM. Next the second stage on both the DA and RAM was not the greatest in design and frankly a lot of people don't know how set them up to get the best out of them, The AK uses the HPR which is a upgraded second stages that was originally designed as a replacement for the DA/RAM second stage. It has better flow performance, has a more aggressive venturi and setup is much easier. Adding one to a DA or RAM takes either to another level of performance. When set up correctly a RAM with the new diaphragm and HPR start to come close to the performance of a AK but it stills can not match it due to improvements in the can design. The AK's can was specifically designed to improve flow and take full advantage of the venturi that is built into the second stage. Also, unless the DA or RAM you were diving had the current mouthpiece valves, you were losing some performance there as well....and to a lesser extent same goes for the duckbill. Both of which have been improved in the last few years.
Performance was farther improved with the release of the new DSV, with which the venturi built into the AK (and a DA or RAM with the HPR installed) can be adjusted to increase the performance even more.
I had my AK tested by a major repair facility on their new regulator test setup. They ran it twice because they did not believe the first run. They keep the profiles of all the major regs to compare against, the AK outperformed the top of the line Scubapro and Atomics by a wide margin, the only reg the beat it was the top end Posiden.
With that said it still is a DH reg and has to be positioned properly on the body and dove properly.