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Permanental partition models and Markovian Gibbs structures. (English) Zbl 1302.82071
Summary: We study both time-invariant and time-varying Gibbs distributions for configurations of particles into disjoint clusters. Specifically, we introduce and give some fundamental properties for a class of partition models, called permanental partition models, whose distributions are expressed in terms of the $$\alpha$$-permanent of a similarity matrix parameter. We show that, in the time-invariant case, the permanental partition model is a refinement of the celebrated Pitman-Ewens distribution; whereas, in the time-varying case, the permanental model refines the Ewens cut-and-paste Markov chains [H. Crane, J. Appl. Probab. 48, No. 3, 778–791 (2011; Zbl 1235.60092)]. By a special property of the $$\alpha$$-permanent, the partition function can be computed exactly, allowing us to make several precise statements about this general model, including a characterization of exchangeable and consistent permanental models.
Reviewer: Reviewer (Berlin)

##### MSC:
 82C22 Interacting particle systems in time-dependent statistical mechanics
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##### References:
 [1] Aldous, D.J., Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the Mean-field theory for probabilists, Bernoulli, 5, 3-48, (1999) · Zbl 0930.60096 [2] Arratia, R., Barbour, A., Tavaré, S.: Logarithmic Combinatorial Structures: a Probabilistic Approach. Eur. Math. Soc., Zurich (2003) · Zbl 1040.60001 [3] Berestycki, J., Exchangeable fragmentation-coalescence processes and their equilibrium measures, Electron. J. Probab., 9, 770-824, (2004) · Zbl 1064.60192 [4] Berestycki, N.; Pitman, J., Gibbs distributions for random partitions generated by a fragmentation process, J. Stat. Phys., 127, 381-418, (2007) · Zbl 1126.82013 [5] Bertoin, J., Homogeneous fragmentation processes, Probab. Theory Relat. Fields, 121, 301-318, (2001) · Zbl 0992.60076 [6] Bertoin, J.: Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics, vol. 102. Cambridge University Press, Cambridge (2006) · Zbl 1107.60002 [7] Burke, C.J.; Rosenblatt, M., A Markovian function of a Markov chain, Ann. Math. Stat., 29, 1112-1122, (1958) · Zbl 0100.34402 [8] Crane, H., A consistent Markov partition process generated from the paintbox process, J. Appl. Probab., 43, 778-791, (2011) · Zbl 1235.60092 [9] Crane, H.: Homogeneous cut-and-paste processes. Manuscript (2013) · Zbl 0991.60038 [10] Crane, H., Some algebraic identities for the $$α$$-permanent, Linear Algebra Appl., (2013) · Zbl 1283.15026 [11] Diaconis, P.; Evans, S.N., Immanants and finite point processes, J. Comb. Theory, Ser. A, 91, 305-321, (2000) · Zbl 0965.15007 [12] Donnelly, P.; Grimmett, G., On the asymptotic distribution of large prime factors, J. Lond. Math. Soc., 47, 395-404, (1993) · Zbl 0839.11039 [13] Efron, B.; Thisted, R., Estimating the number of unseen species: how many words did shakespeare know?, Biometrika, 63, 435-447, (1976) · Zbl 0344.62088 [14] Ewens, W.J., The sampling theory of selectively neutral alleles, Theor. Popul. Biol., 3, 87-112, (1972) · Zbl 0245.92009 [15] Fisher, R.; Corbet, A.; Williams, C., The relation between the number of species and the number of individuals in a random sample of an animal population, J. Anim. Ecol., 12, 42-58, (1943) [16] Fulton, W., Harris, J.: Representation Theory: a First Course. Graduate Texts in Mathematics/Readings in Mathematics. Springer, Berlin (1991) · Zbl 0744.22001 [17] Gyires, B., Discrete distribution and permanents, Publ. Math. (Debr.), 20, 93-106, (1973) · Zbl 0293.15009 [18] Hartigan, J.A., Partition models, Commun. Stat., Theory Methods, 19, 2745-2756, (1990) [19] Hough, J.B.; Krishnapur, M.; Peres, Y.; Virág, B., Determinantal processes and independence, Probab. Surv., 3, 206-229, (2006) · Zbl 1189.60101 [20] Hughes, B.D.: Random Walks and Random Enviroments. Oxford University Press, London (1996) [21] Jerrum, M., Sinclair, A., Vigoda, E.: A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries. J. ACM 671-697 (2004) · Zbl 1204.65044 [22] Kelly, F.: Reversibility in Stochastic Networks. Wiley, New York (1979) · Zbl 0422.60001 [23] Kingman, J.F.C., The representation of partition structures, J. Lond. Math. Soc., 18, 374-380, (1978) · Zbl 0415.92009 [24] Kingman, J.F.C.: Mathematics of Genetic Diversity. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 34. SIAM, Philadelphia (1980) · Zbl 0458.92009 [25] Kingman, J.F.C., The coalescent, Stoch. Process. Appl., 13, 235-248, (1982) · Zbl 0491.60076 [26] Kou, S.C.; McCullagh, P., Approximating the $$α$$-permanent, Biometrika, 96, 635-644, (2009) · Zbl 1206.62144 [27] McCullagh, P., What is a statistical model?, Ann. Stat., 30, 1225-1310, (2002) · Zbl 1039.62003 [28] McCullagh, P.: An asymptotic approximation for the permanent of a doubly stochastic matrix. J. Stat. Comput. Simul. 1-11 (2012) · Zbl 1189.60101 [29] McCullagh, P.; Møller, J., The permanental process, Adv. Appl. Probab., 38, 873-888, (2006) · Zbl 1126.60040 [30] McCullagh, P.; Yang, J., How many clusters?, Bayesian Anal., 3, 101-120, (2008) · Zbl 1330.62033 [31] Minc, H.: Permanents. Addison-Wesley, Reading (1978) [32] Pitman, J., Exchangeable and partially exchangeable random partitions, Probab. Theory Relat. Fields, 102, 145-158, (1995) · Zbl 0821.60047 [33] Pitman, J., Combinatorial stochastic processes, July 7-24 (2002), Berlin [34] Rinaldo, A.; Shalizi, C.R., Consistency under subsampling of exponential random graph models, Ann. Stat., 41, 508-535, (2013) · Zbl 1292.62052 [35] Soshnikov, A., Determinantal random point fields, Russ. Math. Surv., 55, 923-975, (2000) · Zbl 0991.60038 [36] Valiant, L.G., The complexity of computing the permanent, Theor. Comput. Sci., 8, 189-201, (1979) · Zbl 0415.68008 [37] Vere-Jones, D., Alpha-permanents and their applications to mulivariate gamma, negative binomial and ordinary binomial distributions, N.Z. J. Math., 26, 125-149, (1997) · Zbl 0879.15003 [38] Vere-Jones, D., A generalization of permanents and determinants, Linear Algebra Appl., 111, 119-124, (1998) · Zbl 0665.15007
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