The Urn of Hill, Lane and Sudderth

TL;DR

This work analyzes the Hill–Lane–Sudderth (HLS) urn, a memory-bearing reinforced stochastic process with capacity $T$, where the probability of adding a black ball is given by $\pi(\psi_n)$ depending on the current urn share $\psi_n$. It develops a unified framework combining continuous embedding, thermodynamic scaling, and path-integral/Lattice Field Theory methods to characterize typical trajectories and fluctuations, including zero-cost trajectories and entropy/MGF relations. Key contributions include the explicit construction of the transformed urn function $\Pi$ for trajectory reconstruction, a variational large-deviation principle for event probabilities, and a lattice-field-theory perspective that links probability with field-theoretic formalisms via Varadhan and Mogulskii theorems. The results connect probability, statistical mechanics, and models of increasing returns, offering a rigorous basis for studying reinforced dynamics and their applications, while leaving open the exact integration of nonlinear equations for the MG function and entropy density.

Abstract

We review some facts, properties and applications of the urn of Hill, Lane and Sudderth, a paradigmatic model of stochastic process with memory where the urn evolution is as follows: consider an urn of given capacity, at each step a new ball, black or white, is added to the urn with probability that is function (urn function) of the fraction of black balls. The process runs until capacity is reached.

Figures (9)

• Figure 1: Scheme for the step of the HLS model. The urn evolution is as follows: for each step a new ball is added, whose color depends on of the fraction of black balls $\psi$ according to the functional order parameter $\pi$ (urn function).
• Figure 2: On the left: example of urn function, from Franchini 2017 Franchini_URNS. On the right: examples of zero--cost trajectories for the same urn function.
• Figure 3: An influential theory for Increasing Returns was proposed by W. B. Arthur to explain the lock--in phenomenon AEKFBArthNat. In the most simplified situation two competing products gain customers according to a majority mechanism: each new customer asks which product was chosen by a certain (odd) number of previous customers, then buy the most shared product within this sample (at least three). It is known that one of these two products reaches monopoly almost surely in the limit of infinite customers. In figure: Scheme for the step of the IRT model.
• Figure 4: From IRT to HLS.
• Figure 5: Urn function and optimal trajectories for the IRT model. In blue some simulated trajectories from Dosi et al. 2018 DMS