Timeline for Distribution of 1-norm for Gaussian Unitary Ensemble
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
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| Jan 31, 2010 at 10:11 | history | edited | Yemon Choi | CC BY-SA 2.5 |
tidied up the LaTeX
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| Oct 27, 2009 at 1:47 | comment | added | Steve Flammia | Wow, that's great! I can get the upper bound from here; I already estimated that one using 2-norm and Cauchy-Schwarz. I didn't realize that it was tight because my numerics was off, so I was dreaming about an impossible bound. Thanks for your help. | |
| Oct 27, 2009 at 1:02 | vote | accept | Steve Flammia | ||
| Oct 27, 2009 at 13:53 | |||||
| Oct 26, 2009 at 22:43 | history | edited | Yemon Choi | CC BY-SA 2.5 |
added draft effort at solution
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| Oct 26, 2009 at 22:22 | comment | added | Steve Flammia | Oh, and I should add: I'm mostly interested in an upper bound. So don't worry too much about the lower bound. | |
| Oct 26, 2009 at 22:09 | comment | added | Steve Flammia | Yes, that's helpful, thank you. Do I understand that correctly, that this implies the optimal scaling I can hope for with Z is n^{3/2}? I did a bit of numerics, and (at least for small n) it seemed like Z = O(n) up to maybe some log factors. (I'm ignoring a constant that depends on \sigma...) So this is a bit surprising. | |
| Oct 26, 2009 at 20:53 | comment | added | Yemon Choi | OK, I can almost (modulo some CLT handwaving) show that, for any c\in (0,1/2), \sigma c \leq \liminf_n Z n^{-3/2} \leq \limsup_n Z n^{-3/2} \leq \sigma The upper bound is, as you allude to, just the 2-norm and Cauchy-Schwarz. The lower one would follow from some known upper bounds on the spectral radius in the GUE (and I have a reasonably elementary proof of the analogous sp. rad. bound for the GOE, I don't think the complex case is significantly different). Does this sound helpful? If so, I'll try to tighten up the argument - might post it on my blog if it checks out but looks a bit long. | |
| Oct 26, 2009 at 20:01 | comment | added | Steve Flammia | As long as the estimate isn't too crude, then it should be fine. There is one obvious way to get an estimate, which is to use a bound on the 1-norm in terms of the 2-norm, picking up a dimensional factor along the way. I'd like to avoid that, if possible. | |
| Oct 26, 2009 at 19:43 | history | answered | Yemon Choi | CC BY-SA 2.5 |