Abstract
We establish the soundness of the replica symmetric ansatz introduced by M. Mézard and G. Parisi for the minimum matching problem in the pseudo-dimension $d$ mean field model for $d\geq 1$. The case $d=1$ corresponds to the $\pi^2/6$-limit for the assignment problem proved by D. Aldous in 2001.
We introduce a game-theoretical framework by which we establish the analogous limit also for $d>1$.
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@article {A92, MRKEY = {1183889},
AUTHOR = {Aldous, David J.},
TITLE = {Asymptotics in the random assignment problem},
JOURNAL = {Probab. Theory Related Fields},
FJOURNAL = {Probability Theory and Related Fields},
VOLUME = {93},
YEAR = {1992},
NUMBER = {4},
PAGES = {507--534},
ISSN = {0178-8051},
CODEN = {PTRFEU},
MRCLASS = {60C05 (05C70 05C80)},
MRNUMBER = {1183889},
MRREVIEWER = {Mark R. Jerrum},
DOI = {10.1007/BF01192719},
ZBLNUMBER = {0767.60006},
} -
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MRNUMBER = {1839499},
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DOI = {10.1002/rsa.1015},
ZBLNUMBER = {0993.60018},
} -
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AUTHOR = {Aldous, David J. and Bandyopadhyay, Antar},
TITLE = {A survey of max-type recursive distributional equations},
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MRNUMBER = {2134098},
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DOI = {10.1214/105051605000000142},
ZBLNUMBER = {1105.60012},
} -
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} -
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ZBLNUMBER = {0992.82500},
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AUTHOR = {Parisi, G.},
TITLE = {Spin glasses and optimization problems without replicas},
BOOKTITLE = {Le Hasard et la Matière/Chance and Matter},
VENUE={{L}es {H}ouches, 1986},
EDITOR={Souletie, J. and Vannimenus, J. and Stora, R.},
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PUBLISHER = {Elsevier Science Publishers B. V.},
ADDRESS = {Netherlands},
YEAR = {1987},
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author={Salez, Justin and Shah, Devavrat},
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