arXiv:0805.0113v1 [hep-th] 1 May 2008 New Strings for Old Veneziano Amplitudes IV. Connections With Spin Chains and Other Stochastic Systems Arkady Kholodenko 375 H.L.Hunter Laboratories, Clemson University, Clemson, SC 29634-0973, U.S.A. E-mail: string@clemson.edu Abstract: In a series of recently published papers we reanalyzed the exist- ing treatments of the Veneziano and Veneziano-like amplitudes and the models associated with these amplitudes. In this work we demonstrate that the already obtained new partition function for these amplitudes can be exactly mapped into that for the Polychronakos-Frahm (P-F) spin chain model which, in turn, is obtainable from the Richardon-Gaudin (R-G) XXX model. Reshetikhin and Varchenko demonstrated that such a model is obtainable as a leading approxi- mation in their WKB-type analysis of solutions of the Knizhnik-Zamolodchikov (K-Z) equations. The linear independence of solutions of these equations is con- trolled by determinants (discovered by Varchenko) whose explicit form up to a constant coincides with the Veneziano (or Veneziano-like) amplitudes. In the simplest case, when K-Z equations are reducible to the Gauss hypergeo- metric equation, the determinantal conditions coincide with those which were discovered by Kummer in 19-th century. Kummer’s results admit physical in- terpretation crucial for providing needed justification associating determinantal formula(s) with Veneziano-like amplitudes. General results are illustrated by many examples. These include but are not limited to only high energy physics since all high energy physics scattering processes can be looked upon from much broader stochastic theory of random fragmentation and coagulation processes recently undergoing active development in view of its applications in disciplines ranging from ordering in spin glasses and population genetics to computer sci- ence, linguistics and economics, etc. In this theory Veneziano amplitudes play a central (universal) role since they are the Poisson-Dirichlet-type distributions for these processes (analogous to the more familiar Maxwell distribution for gases). Keywords: Polychronakos and Richardson-Gaudin spin chains, Knizhnik- Zamolodchikov equations, determinantal formulas, Veneziano amplitudes, ran- dom fragmentation-coagulation processes. 1