In the previous post we described a fairly straightforward argument, using generating functions and the saddle-point bound, for giving an upper bound
Image may be NSFW.
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on the partition function Image may be NSFW.
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. In this post I’d like to record an elementary argument, making no use of generating functions, giving a lower bound of the form Image may be NSFW.
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for some Image may be NSFW.
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, which might help explain intuitively why this exponential-of-a-square-root rate of growth makes sense.
The starting point is to think of a partition of Image may be NSFW.
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as a Young diagram of size Image may be NSFW.
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Clik here to view.. Heuristically, if the path takes a total of Image may be NSFW.
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steps then there are about Image may be NSFW.
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such paths, and if the area under the path is Image may be NSFW.
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, so this already goes a long way towards explaining the exponential-of-a-square-root behavior.
We can make this argument into a rigorous lower bound as follows. Consider lattice paths beginning at Image may be NSFW.
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and ending at Image may be NSFW.
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where Image may be NSFW.
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is a positive integer to be determined later. Suppose that the steps of the lattice paths alternate between paths of the form (down, right, right, down) and (right, down, down, right), which means that Image may be NSFW.
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, and the number of such paths is exactly Image may be NSFW.
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. This gives
Image may be NSFW.
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whenever Image may be NSFW.
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Clik here to view. where Image may be NSFW.
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, quite a bit worse than the correct value Image may be NSFW.
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. This bound generalizes to all values of Image may be NSFW.
Clik here to view. with only a small loss in the exponent because Image may be NSFW.
Clik here to view. is nondecreasing (since the lattice path can continue along the line Image may be NSFW.
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for awhile at the end before hitting the x-axis).
One reason this construction can’t produce a very good bound is that the partitions we get this way do not resemble the “typical” partition, which (as proven by Vershik and explained by David Speyer here) is a suitably scaled version of the curve
Image may be NSFW.
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.
whereas our partitions resemble the curve Image may be NSFW.
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. With a more convex curve we can afford to make the path longer while fixing the area under it.
So let’s remove the restriction that our curve resemble Image may be NSFW.
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Clik here to view. directly, we will count Image may be NSFW.
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, so the number of lattice paths with area at most Image may be NSFW.
Clik here to view.. Since Image may be NSFW.
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times this count. And we have much more freedom to pick a path now that we only need to bound its area rather than find it exactly. We can now take the path to be any Dyck path from Image may be NSFW.
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Clik here to view., of which there are
Image may be NSFW.
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where Image may be NSFW.
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denotes the Catalan numbers and the asymptotic can be derived from Stirling’s approximation. The area under a Dyck path is at most Image may be NSFW.
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Image may be NSFW.
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and hence, when Image may be NSFW.
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),
Image may be NSFW.
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which (ignoring polynomial factors) is of the from Image may be NSFW.
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where Image may be NSFW.
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, a substantial improvement over the previous bound. Although we are now successfully in a regime where our counts include paths of a typical shape, we’re overestimating the area under them, so the bound is still not as good as it could be.