Linearly edge-reinforced random walks

We review results on linearly edge-reinforced random walks. On finite graphs, the process has the same distribution as a mixture of reversible Markov chains. This has applications in Bayesian statistics and it has been used in studying the random walk on infinite graphs. On trees, one has a representation as a random walk in an independent random environment. We review recent results for the random walk on ladders: recurrence, a representation as a random walk in a random environment, and estimates for the position of the random walker.Comment: Published at http://dx.doi.org/10.1214/074921706000000103 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

Simple Amplitudes for \Phi^3 Feynman Ladder Graphs

Recently, we proposed a new approach for calculating Feynman graphs amplitude using the Gaussian representation for propagators which was proven to be exact in the limit of graphs having an infinite number of loops. Regge behavior was also found in a completely new way and the leading Regge trajectory calculated. Here we present symmetry arguments justifying the simple form used for the polynomials in the Feynman parameters $\bar \alpha _{\ell}$, where $\bar \alpha _{\ell}$ is the mean-value for these parameters, appearing in the amplitude for the ladder graphs. (Taking mean-values is equivalent to the Gaussian representation for propagators).Comment: 11 Plain TeX pages, 2 PostScript figures include

Three-State Complex Valued Spins Coupled to Binary Branched Polymers in Two-Dimensional Quantum Gravity

A model of complex spins (corresponding to a non-minimal model in the language of CFT) coupled to the binary branched polymer sector of quantum gravity is considered. We show that this leads to new behaviour.Comment: 3 pages, Latex2e, 2 eps figures, uses espcrc2 and epsf. Contribution to LATTICE 97, to appear in the Proceeding

Branched polymers, complex spins and the freezing transition

We show that by coupling complex three-state systems to branched-polymer like ensembles we can obtain models with gamma-string different from one half. It is also possible to study the interpolation between dynamical and crystalline graphs for these models; we find that only when geometry fluctuations are completely forbidden is there a crystalline phase.Comment: 14 pages plain LateX2e, 4 eps figures included using eps

Regge behaviour and Regge trajectory for ladder graphs in scalar Φ3\Phi^3 field theory

Using the gaussian representation for propagators (which can be proved to be exact in the infinite number of loops limit) we are able to derive the Regge behaviour for ladder graphs of $\phi^3$ field theory in a completely new way. An analytic expression for the Regge trajectory $\alpha (t/m^2)$ is found in terms of the mean-values of the Feynman $\alpha$-parameters. $\alpha (t/m^2)$ is calculated in the range $- 3.6 < t/m^2 < 0.8$. The intercept $\alpha (0)$ agrees with that obtained from earlier calculations using the Bethe-Salpeter approach for \alpha (0) \gsim 0.3.Comment: 10 PlainTex pages, 2 PostScript Figures include