Most laptops are pretty good these days. I've been having good luck with Acer recently, so If it were me I'd probably buy this:
8Gb Ram and an i7 processor should be adequate for most live looping. You could swap out the drive for a SSD (solid state drive) if you have a large number of tracks of music being written to or read from the disk.
The biggest question mark is your sound card. Most people anymore use a USB sound card, and not all chipsets work equally well with USB audio. I would post in whatever forums there are for your particular sound card and find out what laptops have worked for other people. I'm currently using a pretty inexpensive Focusrite Scarlett 2i2 USB sound card and I'm getting low levels of latency that rival expensive RME sound cards.
With a $1000 budget you should be fine. Stick with lots of RAM and a top of the line Intel i7 processor and you will probably never come close to the limits of the machine.