Related papers: Regular Hilberg Processes: An Example of Processes…

We consider a non-conserving zero-range process with hopping rate proportional to the number of particles at each site. Particles are added to the system with a site-dependent creation rate, and vanish with a uniform annihilation rate. On a…

In this article we establish two fundamental results for the sublevel set persistent homology for stationary processes indexed by the positive integers. The first is a strong law of large numbers for the persistence diagram (treated as a…

We construct mixing processes over an infinite alphabet and ergodic processes over a finite alphabet for which Shannon mutual information between adjacent blocks of length $n$ grows as $n^\beta$, where $\beta\in(0,1)$. The processes are a…

We consider non-ergodic class of stationary real harmonizable symmetric $\alpha$-stable processes $X=\left\{X(t):t\in\mathbb{R}\right\}$ with a finite symmetric and absolutely continuous control measure. We refer to its density function as…

This paper addresses the problem of determining the distance between two regular languages. It will show how to expand Jaccard distance, which works on finite sets, to potentially-infinite regular languages. The entropy of a regular…

We consider an infinite extension $K$ of a local field of zero characteristic which is a union of an increasing sequence of finite extensions. $K$ is equipped with an inductive limit topology; its conjugate $\bar{K}$ is a completion of $K$…

We study the multiplicative version of the classical Furstenberg's filtering problem, where instead of the sum $\mathbf{X}+\mathbf{Y}$ one considers the product $\mathbf{X}\cdot \mathbf{Y}$ ($\mathbf{X}$ and $\mathbf{Y}$ are bilateral,…

We introduce a class of (2+1)-dimensional stochastic growth processes, that can be seen as irreversible random dynamics of discrete interfaces. "Irreversible" means that the interface has an average non-zero drift. Interface configurations…

Reinforcement Learning (RL) has proven highly effective at enhancing the complex reasoning abilities of Large Language Models (LLMs), yet underlying mechanisms driving this success remain largely opaque. Our analysis reveals that puzzling…

Existence and stability properties are studied for Hawkes process, i.e. point process $S$ that has long-memory and intensity $r(t)=\lambda \big(g_0(t)+ \sum_{\tau<t, \tau \in S} h(t-\tau) \big)$. The approach to Hawkes process presented in…

Text generation tasks, including translation, summarization, language models, and etc. see rapid growth during recent years. Despite the remarkable achievements, the repetition problem has been observed in nearly all text generation models…

Consider a uniformly sampled random $d$-regular graph on $n$ vertices. If $d$ is fixed and $n$ goes to $\infty$ then we can relate typical (large probability) properties of such random graph to a family of invariant random processes (called…

An \emph{evolving Shelah-Spencer process} is one by which a random graph grows, with at each time $\tau \in {\bf N}$ a new node incorporated and attached to each previous node with probability $\tau^{-\alpha}$, where $\alpha \in (0,1)…

We introduce Gradient Flow Aggregation (GFA), a random growth model. Given a set of existing particles $\left\{x_1, \dots, x_n\right\} \subset \mathbb{R}^2$, a new particle arrives from a random direction at $\infty$ and flows in direction…

We consider statistical-mechanical models for spin systems built on hierarchical structures, which provide a simple example of non-mean-field framework. We show that the coupling decay with spin distance can give rise to peculiar features…

Hilberg (1990) supposed that finite-order excess entropy of a random human text is proportional to the square root of the text length. Assuming that Hilberg's hypothesis is true, we derive Guiraud's law, which states that the number of word…

In this paper we introduce and exploit the real replica approach for a minimal generalization of the Hopfield model, by assuming the learned patterns to be distributed accordingly to a standard unit Gaussian. We consider the high storage…

In the first chapter of Shannon's "A Mathematical Theory of Communication," it is shown that the maximum entropy rate of an input process of a constrained system is limited by the combinatorial capacity of the system. Shannon considers…

Runtime repeated recursion unfolding was recently introduced as a just-in-time program transformation strategy that can achieve super-linear speedup. So far, the method was restricted to single linear direct recursive rules in the…

We analyse the entanglement structure of states generated by random constant-depth two-dimensional quantum circuits, followed by projective measurements of a subset of sites. By deriving a rigorous lower bound on the average entanglement…