I am working on a project to control a 64 x 64 LED matrix via Bluetooth.
The 64 x 64 LED matrix requires a 5V 4A power supply. Ideally I would like to power the esp32 from the same source.
From research I can use a buck converter to do this, is this correct and is there any research I can do on the cable size. If I power an ESP from a 9V battery I've always used small protoytpe jumper cables but I'm thinking if I do this from a larger power source they'll get pretty hot pretty quickly.
You are misunderstanding some basic facts of electrical circuits. You can attach a 100 AMP power supply to that esp32, the board will still only draw the few milliamps it needs. Current (generally) is 'pulled' from the supply. Do you know OHMS law, understand it? SInce I = V/R, if V and R are unchanged, then I is unchanged regardless of the ability of the PSU. I will ONLY increase if the Voltage increases, or the Resistance of the load decreases. Neither is changing here.
Yes that will work fine but it may not be needed depending on your Arduino. Also set the output voltage of the buck converter before connecting it to your Arduino. There are wire tables on line where you can check wire size. AWG - American Wire Gauge Current Ratings There are two main items, voltage and current. Voltage basicly references the insulation and current the conductor size. Any convenient size wire such as 22 AWG would be fine. The only time it would get hot is if it were shorted or overloaded with something.