Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · math-ph/0610026

Asymptotic Feynman-Kac formulae for large symmetrised systems of random walks

Stefan Adams·Tony Dorlas
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study large deviations principles for $ N $ random processes on the lattice $ \Z^d $ with finite time horizon $ [0,β] $ under a symmetrised measure where all initial and terminal points are uniformly given by a random permutation. That is, given a permutation $ σ$ of $ N $ elements and a vector $ (x_1,...,x_N) $ of $ N $ initial points we let the random processes terminate in the points $ (x_{σ(1)},...,x_{σ(N)}) $ and then sum over all possible permutations and initial points, weighted with an initial distribution. There is a two-level random mechanism and we prove two-level large deviations principles for the mean of empirical path measures, for the mean of paths and for the mean of occupation local times under this symmetrised measure. The symmetrised measure cannot be written as any product of single random process distributions. We show a couple of important applications of these results in quantum statistical mechanics using the Feynman-Kac formulae representing traces of certain trace class operators. In particular we prove a non-commutative Varadhan Lemma for quantum spin systems with Bose-Einstein statistics and mean field interactions. A special case of our large deviations principle for the mean of occupation local times of $ N $ simple random walks has the Donsker-Varadhan rate function as the rate function for the limit $ N\to\infty $ but for finite time $ β$. We give an interpretation in quantum statistical mechanics for this surprising result.

Explore related subjects

math-phmath.MPmath.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Stefan Adams, Tony Dorlas. 2006-10-11. Asymptotic Feynman-Kac formulae for large symmetrised systems of random walks. https://arxiv.org/abs/math-ph/0610026

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Why we should condition denoising diffusion generative models on windows of past observations

Data assimilation (DA) is, traditionally, a cycling process that relies on time-dependent priors to propagate information from past observations to future cycles. Using denoising diffusion generative modeling for DA is challenging because standard approaches use a fixed training data set, which in turn leads to a static prior that ignores information from past observations. Because past observations are ignored, DA systems with static priors lead to larger posterior errors than cycling DA systems. Incorporating time-dependent priors into generative models, however, requires expensive and frequent retraining. Motivated by linear systems theory - where the dependence of a prediction of a Kalman filter on past observations decays exponentially - we condition diffusion models on short windows of past observations. Specifically, we describe training procedures for two frameworks: a diffusion DA system predicting the current state given a set of past observations, and a diffusion ``direct observation prediction'' (DOP) system, predicting future observations given a set of past observations. Using a canonical linear system, we show that both systems can achieve the minimal posterior error characteristic of a fully-cycled DA/DOP system, without re-training, provided the time windows are long enough. The linear setup ensures analytical tractability, avoids confounding neural network training errors, and confirms that conditioning on windows of past observations is required for efficient and accurate diffusion-based DA or DOP.

math-ph↗
arXiv2026

The kinematic structures and the inertial geometry of a moving charge

We ask how much of the geometry a charged particle moves in is fixed by its motion, and how much a particle must bring. We ask of a symplectic structure only that it relate velocity to momentum as Hamilton's equations do, and we ask it of every energy at once. In particular, we show that the structures meeting that demand are the canonical one and its twists by a closed two-form of the base. A field provides the two-form, a particle the multiplier before it, which we identify constitutively with its charge. Thus, a single energy governs a family of structures, and each particle takes the one its charge fixes. We then ask what a particle must bring to be given a momentum, and we show that the degree of that map settles the degree at which a field enters Newton's Second Law. An antisymmetric bilinear form returns no Lorentz force, whilst a Randers metric returns one --- a length whose difference from a Riemannian one is linear in the velocity. Moreover, we find that metric already within the twisted structure, as its primitive over a level of the free energy, and its law of transport to be nonlinear, no affine connection being known to serve. Under an indefinite signature the length parts from the dynamics, and the extremals turn from shortest to longest. On the round sphere a monopole flux leaves no such metric, whilst the transport remains and prequantisation, given a unit of action, restricts the charge to a lattice. In this manner, we conclude that each charge-to-mass ratio receives a geometry of its own, so that by a functionalist criterion none of them is the spacetime of a charged particle.

math-ph↗