We study large deviations for rare clusters in sparse systems of moving particles. In the regime \(nr_n^d\to0\) and \(\rho_{k,n}=n^kr_n^{d(k-1)}\to\infty\), we prove a large deviation principle for the empirical measure of isolated \(k\)-particle trajectory clusters. The speed is \(\rho_{k,n}\), and the rate function is the relative entropy \(h(\,\cdot\,\mid\tau_k)\), where the finite reference measure \(\tau_k\) explicitly incorporates the underlying path law. As consequences, we derive free-energy variational formulas for bounded interactions and a hard-core constraint and identify the corresponding minimizing cluster law. The normalized optimizer provides the basis for Metropolis--Hastings sampling of interacting trajectory clusters. As an application motivated by chain formation and swarming in active particle systems, we calibrate the resulting stochastic model to experimental trajectories of magnetic micromotors and find that the fitted velocity scale varies systematically with particle size and magnetic forcing.
In this paper, we establish a small-time annealed path large deviation principle for one-dimensional diffusions in a random environment associated with the generator ${\mathcal L}_W f(x)=e^{-\rho(x,W)}(e^{a(x,W)}f'(x))'$. The coefficients $\{\rho(x,\cdot):x\in\mathbb R\}$ and $\{a(x,\cdot):x\in\mathbb R\}$ are random....
Let \(\{Z_n\}_{n\geq0}\) be a supercritical branching random walk with deterministic rooted \(b\)-ary tree, \(b\geq2\), and symmetric displacements satisfying \(\lim_{x\to+\infty}x^{-\alpha}\log\Pp(X>x)=-\lambda \) with \(\alpha,\lambda>0\). Set $\overline Z_n(\cdot):=Z_n(\cdot)/Z_n(\mathbb{R}),~n\geq0.$ For a finite u...
Let \(\alpha, \beta \in \mathbb{T} = \mathbb{R}/\mathbb{Z}\), and let \((\xi_n)_{n \ge 1}\) be independent random variables taking the values \(\alpha\) and \(\beta\) with equal probability. We study the random walk \[X_n = \xi_1 + \cdots + \xi_n \pmod 1\] and present a deterministic Diophantine condition on \((\alpha,...
Let \(\{X_n:n\ge0\}\) be a controlled branching process with controls \(\phi_n\), and let \(N_n=\phi_n(X_n)\) be its progenitor count. We study large deviations of \(X_{n+1}/N_n\), \(N_n/X_n\), and \(X_{n+1}/X_n\), on their natural positive-denominator events, in supercritical and critical regimes. We treat generation-...
Miguel González, C. Minuesa, I. D. del Puerto et al.· 0 citations
We study the eigenvector-weighted spectral statistics \[ T_n(\varphi,g)=p^{-1}tr\{\varphi(\boldsymbol{S}_n)g(\boldsymbol{\Sigma})\} \] of large-dimensional sample covariance matrices \(\boldsymbol{S}_n\) with population covariance \(\boldsymbol{\Sigma}\). These statistics combine functions of sample eigenvalues with sa...
We introduce a sampling approach for energy- and score-based generative models that requires no gradient evaluations of the model. Replacing the drift term that would normally contain the score $\nabla_\mathbf{x} \log p_\theta(\bf{x})$ with a high-frequency dithered cosine of the model's \textit{value}, $\sqrt{\alpha\o...
A. Scheinker· 0 citations
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