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--- talk:people:romik:osp [2019/10/19 18:35]
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+++ talk:people:romik:osp [2019/10/25 19:10] (current)
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-== Comments about the FPSAC draft (added by Dan 10/9/19, comments by Dan and Fabio added later): ==
+== Comments about the FPSAC draft: ==
-Page/line numbers refer to the version of 10/14/19.
+Page/line numbers refer to the version submitted to FPSAC on Oct 25 2019. Add any typo/correction ideas below.
 **Pending comments:**
-  - Figure 2: are we happy with the choice of colors? \\ \\ [E: It seems to me that the colouring in Figure 2 (tables on sorting networks) is a bit misleading. The colouring used for the parameters of the wiring diagrams is the same (or very very very similar, maybe?) to the colouring used for the wiring diagrams themselves. Do you guys confirm, as I think, that they should not be related? If so, I think it’d be better to use different colours for the parameters — as in Example 2, where colours are different from those in Figure 1. I’m aware that this way Figure 1 might thus result in a harlequinade… the current solution is visually more attractive, but it might generate confusion.] \\ \\ [D: I replaced Figure 2 with a new version in which the wiring diagrams are in black and white, that will avoid the confusion and I think the wiring diagram colors aren't as essential for such small examples. Any thoughts? (The older version with the colors is still there so you can revert back if you don't like this.)] \\
-  - Page 6: lemma 1: we haven't defined the RSK and Bur mappings in the context of tableaux with real entries, so this claim (and the sketched proof idea) doesn't make sense for exponential r.v.s. One solution would be to phrase this lemma only for geometric r.v.s and transfer the limiting argument involving geometrics with parameter $p=e^{-\epsilon \alpha}$ as $\epsilon\searrow 0$ down to the proof of Theorem 4. The other alternative would be to expand our treatment of RSK and Bur to the more general setting of tableaux with real entries, but that would perhaps take up more space than can afford. \\ \\ [E: I've adopted the second solution, as it doesn't take more space. To do so, I've generalised Theorem 5, allowing tableaux with entries in a submonoid $M$ of $\mathbb{R}_{\geq 0}$. In the case of geometric weights (geometric starting at 0), $M= \mathbb{Z}_{\geq 0}$; in the case of exponential weights, $M=\mathbb{R}_{\geq 0}$. In this paper https://arxiv.org/abs/1905.09756, I also used $M=\frac{1}{2} \mathbb{Z}_{\geq 0}$. The eccentric requirement that $M$ is a submonoid of $\mathbb{R}_{\geq 0}$ is motivated as follows: $M$ must contain $0$, otherwise the image set for RSK and Bur won't have such a nice form (i.e. interlacing tableaux with values in the //same// set $M$) -- see also discussion above about the "WLOG" sentence. Also, $M$ must be closed under addition, otherwise you might have input entries in $M$ and output entries outside of $M$. If you reckon that all this is too technical and would require a discussion (which we cannot afford in term of space), I can instead adopt the first solution proposed by Dan and postpone these technicalities to the full version.] \\ \\ [D: I think this is okay, though the mention of submonoids is a bit distracting and does add an extra layer of technicality that doesn't seem completely necessary. Here is another suggestion: maybe in Theorem 5, instead of "let $(M,+)$ be a submonoid of $(\mathbb{R}_{\ge 0},+)$ we can say "... and  let $M$ be one of the sets $\mathbb{Z}_{\ge 0}$ or $\mathbb{R}_{\ge 0}$. There exist two bijections ..."? Then we won't need to mention submonoids (though if you want we can add a comment in the full version that the result extends to submonoids).] \\
-  - Page 9, line -1: the expression $\frac{F_n(x_1,\ldots,x_{n-1})}{G_n(x_1,\ldots,x_{n-1})}$ doesn't make sense, fix this remark. \\ \\ [E: I had also noticed this. However, it's not totally clear to me what the limit procedure should be. Can somebody else fix this?] \\ \\ [D: I reverted back to something close to one of your earlier attempts to clarify this, which I understand now was indeed better. Please check.] \\
-  - E: How do we denote the set of non-negative reals, now that we need it for Theorem 5 (provided that we keep the generalisation I've introduced)? One option would be $\mathbb{R}_+$, but it's ambiguous because it's not clear if $0$ is included. I personally like the notation $\mathbb{Z}_{\geq 0}$ and $\mathbb{R}_{\geq 0}$ because it's not ambiguous and it's the same for integers and reals. Anyway, I've set up macros \Znonneg and \Rnonneg for non-negative integers and reals respectively: feel free to change the notation if you don't like the current one. \\ \\ [D: this is fine, and I agree that having consistent notation is nice.]
-  - E: I suggest that we remove the 6 definitions ‘corner times’, ‘corner permutation’, ‘degree sequence’, ‘last swap times’, ‘order of last swap times’ and again ‘degree sequence’, for three reasons: 1) we never use them, except for maybe 1-2 occurrences of “the degree sequence of t” which can just be replaced with “deg_t”; 2) some of them are pretty verbose, like “order of last swap times”; 3) the definition “last swap times” can be confused with U_n, which is also defined in Section 1 as the vector of “last swap times”. What do you think? \\
-  - E: I’ve replaced a few occurrences of “colour” and “colouring” with “color” and “coloring”, as it seems we’ve been using American spelling throughout the paper. \\ \\ [D: funny, I was trying to be considerate to my Irish coauthors by using the British spelling for colouring, but I guess I wasn't paying attention to other spelling differences... Anyway I'm happy with either British or American.]
+  - No comments at this point.
-**Comments that were addressed and don't require more discussion:
-**
-  - Page 11: "... by the vectors of the discrete finishing times ...": given that we are no longer using the term "finishing times", I suggest deleting this and just saying "the vectors $\overline{\textrm{cor}}_t$ and $\overline{\textrm{last}}_s$". [F: this part is only descriptive and I would keep "discrete finishing time" if this helps the reader. If you think it causes confusion than we can remove.] [D: Yes, I think it will be confusing to a reader who has never heard the term mentioned before except in the different context of "particle finishing times". I removed those terms.] [F: OK.]
-  - Title of the paper: after thinking about it some more, it seems to me that changing the title to "Sorting networks, staircase Young tableaux and last passage percolation" would be a slight improvement, putting the emphasis a bit more on sorting networks. Any thoughts? [F: Yes. This would also agree with the order in which we introduce the processes in the abstract and in the text.] [E: agree, done.]
-  - Pages 5-6, the notations $\textrm{Tab}_{\mathbb{N}_0}(\lambda)$ and $\textrm{IntTab}_{\mathbb{N}_0}(\lambda)$: decide if this is the notation we want to stick with or what should come instead. [E: I'm fine with this notation. BTW, the meaning of the previous notation $\mathbb{Z}^{\lambda}$ was: functions from the boxes of $\lambda$ to $\mathbb{Z}$, i.e. $\lambda$-shaped arrays of integers. But I agree that the hat indicating interlacing tableaux wasn't a great choice.]
-  - Page 7, beginning of section 3.1: are we happy with the "cor" notation? [F: OK for me.] [E: yes]
-  - Page 9, proof sketch of lemma 2: I might try to improve this a bit. [D: added more details and replaced "Proof sketch" with "Proof".]
-  - Finalize the companion Mathematica package and put the final version on the web page I created. [D: done!]
-  - Page 3, line 6: change "$(n-1)$-th" to "$(n-1)$th". [E: done]
-  - Page 3, line -1: change "$\mathbf{U}_n(j) \stackrel{D}{=} \mathbf{V}_n(j)$" to "$\mathbf{U}_n(k) \stackrel{D}{=} \mathbf{V}_n(k)$" (the symbol $k$ is more consistent with the notation we're using elsewhere). [E: done]
-  - Page 4, line 3 of section 1.3: change "which is of independent interest" to "that is of independent interest" (more grammatically correct as some people believe, though I think there isn't a universal consensus on this, so really it's fine either way). [E: done]
-  - Page 5, lines 1-2: change "equivalently if it is the last index of the corresponding diagonal" to "equivalently if $(i,j)$ is the last box of its diagonal". (In the current version, the first "it" is ambiguous since it could refer to $(i,j)$ or to $(i+1,j+1)$, and separately from that it's inaccurate to refer to $(i,j)$ as an "index".) [E: done]
-  - Page 5, line 8: change "random Young tableau" to "random tableau". [E: done]
-  - Page 5, line 9: change "referred to as weights" to "referred to (as before) as weights", or maybe remove this clause. [E: removed.]
-  - Page 5, line -8: change "starting from $(1,i)$" to "starting at $(1,i)$". [E: done]
-  - Page 6, lemma 1: change "If $X$ is a tableau" to "If $X$ is a random tableau" (unless this causes overflow to the next line) [E: done]
-  - Page 11, line 2: change "A key insight is the observation that" to "A key insight is that" [E: done]
-  - Page 7, line -14: change "$\delta_n \setminus \lambda$" to "$\delta_n \setminus \lambda^{(k)}$". [E: done]
-  - Page 6, proof of Lemma 6: we're using geometric random variables that start from 0, which is a bit unconventional. Make sure we are happy with the way this is written. [F: OK. I suggest deleting  "(assuming WLOG... $0$)". ] [D: I think it's better to keep the WLOG comment. Otherwise the reader might worry that the claim may be true for geometrics starting at 0 but not for the more conventional geometric r.v.s] [E: When dealing with LPP using RSK, one usually considers geometric variables starting at 0, because the RSK is done on nonnegative integer matrices/tableaux, so this is the most natural choice. It's not forbidden to use strictly positive inputs, but one should take care of what the image set is. I haven't really checked, but my intuition is that something on the following lines should happen: input tableaux with strictly positive integer entries are bijectively mapped, under RSK, to output tableaux with integers $\geq k$ on the $k$-th antidiagonal. In our case, everything should still work for geometric variables starting at 1, provided that both RSK and Burge correspondences have the //same// image set when restricted to tableaux with strictly positive integer entries (otherwise one can conclude that there exists a tableau $t$ such that $P(\rm{RSK}(X)=t) \neq P(\rm{Bur}(X)=t)$). I believe the latter fact is true, but it should be properly checked. Even if we address this, there would be no space to include a discussion about this in the extended abstract. So I suggest that we postpone the issue to the full version. Since the issue doesn't seem to be so immediate, I would avoid saying that "WLOG we assume...". At least for the extended abstract, I don't think there's any harm in taking for granted (with no further explanation) that our geometric variables start at 0 -- even in the wikipedia page on the geometric distribution, both parametrisations (starting at 0 and starting at 1) are presented as equally "standard". What do you reckon?] [D: Ok, I'm convinced. I removed the WLOG comment.]
-  - E: Start of section 2: Removed sentence about the identification of the boxes of a partition with its lattice vertices, as a similar sentence is already in Section 1 in the paragraph ‘Randomly growing a staircase shape Young diagram’ (now rephrased a bit).
-  - E: Deleted “of this extended abstract” in the last sentence of Subsection 1.1, to avoid repetitions with the previous sentence.
-  - E: Start of Section 3.3: I think it’s slightly inaccurate to say “V-valued generating functions” (where here V, for brevity, is the vector space defined therein). I’ve changed it to “generating functions as elements of V”. [D: good point!]
-  - E: Sentence just after Example 2: specified which examples we refer to: 1 and 2.
-  - E: I’ve rephrased a bit the very last paragraph of the paper to avoid repetitions.
-  - It would be nice to add some more key words (at least "sorting network") if this could be done without overflowing the content to page 13. (I think this can be achieved by adding a few \medskip's before some of the \section and \subsection tags, but maybe this is not a great idea.) [E: agree, added -- without adding \medskip's. There's now more space for keywords in case you wanna add more.] [D: cool, it's fine like this.]
-[D: Deleted to make the page more readable. The old comments can be viewed [[https://numath.com/wiki/doku.php?id=talk:people:romik:osp&rev=1571008858|here]].]
+**Comments that were addressed previously:
+[D: Deleted to clear up space. See the page history for older comments.]
+[D: Earlier comments deleted to make the page more readable. The old comments can be viewed [[https://numath.com/wiki/doku.php?id=talk:people:romik:osp&rev=1571008858|here]].]

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