[Paper Review] Large deviations for empirical path measures in cycles of integer partitions

Stefan Adams|ArXiv.org|Feb 2, 2007

Stochastic processes and statistical mechanics20 references3 citations

TL;DR

This paper establishes a large deviations principle for empirical path measures in systems of $N$ Brownian motions on $\mathbb{R}^d$ with symmetrized initial-terminal conditions, where paths are linked via a random permutation. The key result identifies a phase transition in the large-$N$ limit: for $d \geq 3$ at high density or long time horizon, a positive fraction of paths form infinite-length cycles, interpreted as a path-measure analog of Bose-Einstein condensation.

ABSTRACT

Consider a large system of $N$ Brownian motions in $\mathbb{R}^d$ on some fixed time interval $[0,β]$ with symmetrised initial-terminal condition. That is, for any $i$, the terminal location of the $i$-th motion is affixed to the initial point of the $σ(i)$-th motion, where $σ$ is a uniformly distributed random permutation of $1,...,N$. In this paper, we describe the large-N behaviour of the empirical path measure (the mean of the Dirac measures in the $N$ paths) when $ Λ\uparrow\mathbb{R}^d $ and $ N/|Λ| oρ$. The rate function is given as a variational formula involving a certain entropy functional and a Fenchel-Legendre transform. Depending on the dimension and the density $ ρ$, there is phase transition behaviour for the empirical path measure. For certain parameters (high density, large time horizon) and dimensions $ d\ge 3 $ the empirical path measure is not supported on all paths $ [0,\infty) o\mathbb{R}^d $ which contain a bridge path of any finite multiple of the time horizon $ [0,β] $. For dimensions $ d=1,2 $, and for small densities and small time horizon $ [0,β] $ in dimensions $ d\ge 3$, the empirical path measure is supported on those paths. In the first regime a finite fraction of the motions lives in cycles of infinite length. We outline that this transition leads to an empirical path measure interpretation of {\it Bose-Einstein condensation}, known for systems of Bosons.

Motivation & Objective

• To analyze the large-$N$ behavior of empirical path measures in a system of $N$ Brownian motions on $\mathbb{R}^d$ with symmetrized initial-terminal conditions.
• To derive a large deviations principle for the empirical path measure as $N \to \infty$ and $\Lambda_N \uparrow \mathbb{R}^d$ with $N/|\Lambda_N| \to \rho$.
• To identify phase transition behavior in the empirical path measure depending on dimension $d$, density $\rho$, and time horizon $\beta$.
• To connect the emergence of infinite-length cycles in the system to the phenomenon of Bose-Einstein condensation in quantum statistical mechanics.
• To express the rate function via a variational formula involving entropy and Fenchel-Legendre transforms, linking it to discrete shape measures of integer partitions.

Proposed method

• Define the symmetrized measure $\mathbb{P}_N^{\rm sym}$ as an average over all permutations $\sigma \in \mathfrak{S}_N$ and initial positions $x_i \in \Lambda_N$, with paths constrained to return to permuted endpoints.
• Construct the empirical path measure $L_N = \frac{1}{N} \sum_{i=1}^N \delta_{B^{(i)}}$, representing the average distribution of paths in the system.
• Apply large deviations theory to $L_N$ under the scaling $N/|\Lambda_N| \to \rho$, deriving a rate function via a variational formula.
• Express the rate function as the sum of an entropy term (governing discrete shape measures of integer partitions) and a Fenchel-Legendre transform of a free energy functional.
• Use Feynman-Kac representations to link the path measure to quantum statistical mechanics of Bosons, particularly the canonical partition function.
• Analyze the cycle structure of permutations induced by the path endpoints, relating cycle lengths to the macroscopic occupation of low-momentum states in momentum space.

Experimental results

Research questions

• RQ1 How does the empirical path measure behave in the large-$N$ limit when the system is confined to expanding boxes $\Lambda_N \subset \mathbb{R}^d$ with fixed particle density $\rho$?
• RQ2 Under what conditions on dimension $d$, density $\rho$, and time horizon $\beta$ does the empirical path measure exhibit a phase transition?
• RQ3 What is the connection between the emergence of infinite-length cycles in the path system and Bose-Einstein condensation in quantum systems?
• RQ4 How does the rate function for the empirical path measure decompose into entropy and free energy components?
• RQ5 In what way do the discrete shape measures of integer partitions govern the large-$N$ behavior of the system’s cycle structure?

Key findings

• For $d \geq 3$ and high density or long time horizon $\beta$, the empirical path measure is not supported on all paths and exhibits a positive fraction of paths forming cycles of infinite length.
• In dimensions $d = 1,2$, and for small $\rho$ and small $\beta$ in $d \geq 3$, the empirical path measure is supported on all paths, with no macroscopic cycles.
• The appearance of infinite-length cycles corresponds to a phase transition analogous to Bose-Einstein condensation in quantum systems of Bosons.
• The rate function for the empirical path measure is expressed as a variational formula involving the entropy of discrete shape measures of integer partitions and a Fenchel-Legendre transform.
• The cycle structure of the random permutation $\sigma$ induced by the path endpoints is directly linked to the macroscopic occupation of the zero-momentum state in momentum space.
• The critical threshold for the phase transition depends on the dimension $d$, the density $\rho$, and the time horizon $\beta$, with the transition occurring when the system's free energy favors long cycles.