Out of RAM, with lots of RAM available

I recently downloaded the Mindcrack Season 4 map, and I haven't been able to launch the world. Every time I try launching it, I run out of memory.

However, I don't think this should be the case. Just about everyone else who downloaded this file has run it with 2-4GB of RAM, and here I am, attempting it with 2, 4, and 8, and still failing to run it. I have a fairly impressive PC in my opinion, so there isn't a reason why I should be struggling with this world.

I've even tried turning the world to peaceful with NBT Explorer, and removing item frames via MCEdit.

Plus, it looks like Minecraft is only using around 700MB of memory. Could there be something else limiting Minecraft?

I left the launcher open, so here's what I got from it (hopefully you'll understand more of it than I did, which was just that it was running out of memory:
[23:34:40] [Client thread/INFO]: Setting user: MegaPixelPerfect
[23:34:40] [Client thread/INFO]: (Session ID is token:4e3966994c454967a0b7390838c31931:31b5b0938f9848628ba8fe8c3177d33c)
[23:34:41] [Client thread/INFO]: LWJGL Version: 2.9.1
[23:34:41] [Client thread/INFO]: Reloading ResourceManager: Default
[23:34:42] [Sound Library Loader/INFO]: Starting up SoundSystem...
[23:34:42] [Thread-6/INFO]: Initializing LWJGL OpenAL
[23:34:42] [Thread-6/INFO]: (The LWJGL binding of OpenAL. For more information, see http://www.lwjgl.org)
[23:34:42] [Thread-6/INFO]: OpenAL initialized.
[23:34:42] [Sound Library Loader/INFO]: Sound engine started
[23:34:43] [Client thread/INFO]: Created: 512x512 textures-atlas
[23:35:05] [Server thread/INFO]: Starting integrated minecraft server version 1.8.1
[23:35:05] [Server thread/INFO]: Generating keypair
[23:35:05] [Server thread/INFO]: Preparing start region for level 0
[23:35:05] [Server thread/INFO]: Changing view distance to 12, from 10
[23:35:06] [Server thread/INFO]: MegaPixelPerfect[local:E:da95067c] logged in with entity id 0 at (3.991672582126509, 14.51992270337038, -53.72364737152029)
[23:35:06] [Server thread/INFO]: MegaPixelPerfect joined the game
[23:35:49] [Server thread/INFO]: Saving and pausing game...
[23:35:49] [Server thread/INFO]: Saving chunks for level 'Test World'/Overworld
[23:35:49] [Server thread/INFO]: Saving chunks for level 'Test World'/Nether
[23:35:49] [Server thread/INFO]: Saving chunks for level 'Test World'/The End
[23:35:51] [Server thread/INFO]: Stopping server
[23:35:51] [Server thread/INFO]: Saving players
[23:35:51] [Server thread/INFO]: Saving worlds
[23:35:51] [Server thread/INFO]: Saving chunks for level 'Test World'/Overworld
[23:35:51] [Server thread/INFO]: Saving chunks for level 'Test World'/Nether
[23:35:51] [Server thread/INFO]: Saving chunks for level 'Test World'/The End
[23:35:59] [Server thread/INFO]: Starting integrated minecraft server version 1.8.1
[23:35:59] [Server thread/INFO]: Generating keypair
[23:35:59] [Server thread/INFO]: Preparing start region for level 0
[23:36:16] [Server thread/ERROR]: Encountered an unexpected exception
java.lang.OutOfMemoryError: Java heap space
at java.util.Arrays.copyOfRange(Unknown Source) ~[?:1.8.0_25]
at java.lang.String.<init>(Unknown Source) ~[?:1.8.0_25]
at java.io.DataInputStream.readUTF(Unknown Source) ~[?:1.8.0_25]
at java.io.DataInputStream.readUTF(Unknown Source) ~[?:1.8.0_25]
at fn.b(SourceFile:373) ~[1.8.1.jar:?]
at fn.a(SourceFile:45) ~[1.8.1.jar:?]
at fv.a(SourceFile:50) ~[1.8.1.jar:?]
at fn.a(SourceFile:380) ~[1.8.1.jar:?]
at fn.a(SourceFile:48) ~[1.8.1.jar:?]
at fn.a(SourceFile:380) ~[1.8.1.jar:?]
at fn.a(SourceFile:48) ~[1.8.1.jar:?]
at fn.a(SourceFile:380) ~[1.8.1.jar:?]
at fn.a(SourceFile:48) ~[1.8.1.jar:?]
at fn.a(SourceFile:380) ~[1.8.1.jar:?]
at fn.a(SourceFile:48) ~[1.8.1.jar:?]
at fz.a(SourceFile:106) ~[1.8.1.jar:?]
at fz.a(SourceFile:71) ~[1.8.1.jar:?]
at fz.a(SourceFile:15) ~[1.8.1.jar:?]
at brl.a(SourceFile:43) ~[1.8.1.jar:?]
at aqr.a(SourceFile:2713) ~[1.8.1.jar:?]
at bmk.a(SourceFile:235) ~[1.8.1.jar:?]
at bmk.a(SourceFile:33) ~[1.8.1.jar:?]
at bgr.a(SourceFile:28) ~[1.8.1.jar:?]
at bgt.a(SourceFile:561) ~[1.8.1.jar:?]
at qt.e(SourceFile:133) ~[1.8.1.jar:?]
at qt.c(SourceFile:81) ~[1.8.1.jar:?]
at net.minecraft.server.MinecraftServer.k(SourceFile:314) ~[1.8.1.jar:?]
at cyk.a(SourceFile:111) ~[1.8.1.jar:?]
at cyk.i(SourceFile:127) ~[1.8.1.jar:?]
at net.minecraft.server.MinecraftServer.run(SourceFile:418) [1.8.1.jar:?]
at java.lang.Thread.run(Unknown Source) [?:1.8.0_25]
[23:36:16] [Server thread/ERROR]: This crash report has been saved to: C:\Users\Josh\AppData\Roaming\.minecraft\crash-reports\crash-2015-01-02_23.36.16-server.txt
[23:36:16] [Server thread/INFO]: Stopping server
[23:36:16] [Server thread/INFO]: Saving players
[23:36:16] [Server thread/INFO]: Saving worlds
[23:36:16] [Server thread/INFO]: Saving chunks for level 'world'/Overworld
[23:36:16] [Server thread/INFO]: Saving chunks for level 'world'/Nether
[23:36:16] [Server thread/INFO]: Saving chunks for level 'world'/The End