Unified derivation of the limit shape for multiplicative ensembles of random integer partitions with equiweighted parts12. Issue 2 (18th April 2014) - Bogachev, Leonid V.
Abstract We derive the limit shape of Young diagrams, associated with growing integer partitions, with respect to multiplicative probability measures underpinned by the generating functions of the form <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj23twghz0" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />ℱ(z)=∏ℓ=1∞ℱ0(zℓ)</alternatives> (which entails equal weighting among possible parts <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj23twgj02" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />ℓ∈ℕ</alternatives>). Under mild technical assumptions on the function <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj23twgj1m" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />H0(u)=ln(ℱ0(u))</alternatives>, we show that the limit shape <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj23twgj25" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />ω*(x)</alternatives> exists and is given by the equation <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj23twghb2" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />y=γ−1H0(e−γx)</alternatives>, where <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj23twgh8z" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />γ2=∫01u−1H0(u) du</alternatives>. The wide class of partition measures covered by this result includes (but is not limited to) representatives of the three meta‐types of decomposable combinatorial structures — assemblies, multisets, and selections. Our method is based on the usual randomization and conditioning; to this end, a suitable local limit theorem is proved. The proofs are greatly facilitated by working with the cumulants of sums of the part counts rather than with their moments.Copyright © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 227–266, 2015