Timeline for Probability of zero in a random matrix
Current License: CC BY-SA 3.0
13 events
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| Sep 5, 2013 at 15:51 | history | edited | Pietro Majer | CC BY-SA 3.0 |
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| Jun 5, 2012 at 14:45 | comment | added | Pietro Majer | Thank you for the counterexamples. I guess one should focus to a suitable class of (discrete) "log-concave" functions $f$ that is stable under convolution with the characteristic functions $\delta_{ij}$ (grouped, in case) defined above, and that includes them. It would be nice if the preservation of log-concavity for this class under discrete convolution would follow directly from the analogous property for continuous convolution. (Something like $f*_{discr}\delta= (\tilde f*_{cont}\tilde\delta)_{|\mathbb{Z}n}$). | |
| Jun 5, 2012 at 13:17 | comment | added | David E Speyer | Please don't delete this answer though! There are a lot of definitions of discrete convexity, and I am hopeful that one of them will make this solution work. | |
| Jun 5, 2012 at 13:16 | comment | added | David E Speyer |
A simple counterexample: Let $f$ take the values $\begin{matrix} 1 & \epsilon \\ \epsilon & 1 \end{matrix}$ on $(0,0)$, $(0,1)$, $(1,0)$ and $(1,1)$, and be $0$ elsewhere. Let $g$ be $\begin{matrix} \epsilon & 1 \\ 1 & \epsilon \end{matrix}$. Then $f \ast g$ is $\begin{matrix} \epsilon & 1+\epsilon^2 & \epsilon \\ 1+\epsilon^2 & 4 \epsilon & 1+\epsilon^2 \\ \epsilon & 1+\epsilon^2 & \epsilon \end{matrix}$. If $\epsilon$ is small enough that $4 \epsilon < 1+ \epsilon^2$, then this is not log-concave by the above definition.
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| Jun 5, 2012 at 13:04 | comment | added | Brendan McKay | Consider $\prod_{1\le i\lt j\le n} (1-x_ix_j)^{-1}$ for some odd $n$. The sequence $t_k$ which is the coefficient of $\prod_i x_i^k$ is not log-concave, since it is 0 for odd $k$. So some additional conditions are required. | |
| Jun 5, 2012 at 12:41 | comment | added | Pietro Majer | @Gjergji: this definition: $-\log f$ is the restriction to $\mathbb{Z}^2$ of some convex function $\mathbb{R}^2\to (-\infty,+\infty]$. | |
| Jun 5, 2012 at 11:17 | comment | added | Brendan McKay | It will be great if this proof is correct, there will be many applications. But I don't know the theory it is based on. | |
| Jun 5, 2012 at 10:37 | comment | added | Gjergji Zaimi | What definition of log-concavity are you using for $\mathbb Z^2$ for example? If it's $f_{i,j}^4\geq \prod f_{neighbours}$ then a restriction doesn't have to be log-concave... | |
| Jun 5, 2012 at 10:31 | comment | added | cardinal | Re: discrete convolution: For the 1D case (which may or may not be helpful here): J. Keilson and H. Gerber, Some Results for Discrete Unimodality, J. Amer. Stat. Assoc., vol. 66, no. 334, 386-389. | |
| Jun 5, 2012 at 10:26 | history | edited | Pietro Majer | CC BY-SA 3.0 |
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| Jun 5, 2012 at 10:18 | comment | added | Pietro Majer | This should just follow by restriction, if we know that $\mu$ is log-concave on $\mathbb{N}^{n+m}$. But now I can't find the reference for the fact that discrete convolution preserves log-concavity, like continuous convolution does. | |
| Jun 5, 2012 at 10:10 | comment | added | Gjergji Zaimi | Why is $\mu(ku,ku)$ log-concave? | |
| Jun 5, 2012 at 8:42 | history | answered | Pietro Majer | CC BY-SA 3.0 |