Tootfinder

Measure and integration
Teo Banica
https://arxiv.org/abs/2506.15534 https://arxiv.org/pdf/2506.15534…

Measure and integration
This is an introduction to measure theory, integration and function spaces, with all the needed preliminaries included, and with some applications included as well. We first discuss some basic motivations, coming from discrete probability, that we develop in detail, as a preliminary to general measure theory. Then we discuss measure theory, integration and function spaces, all developed in a standard way, and with emphasis on the explicit computation of various integrals. Finally, we come back …

Understanding the rift between update rules in Evolutionary Graph Theory: The intrinsic death rate drives star graphs from amplifying to suppressing natural selection
Max Dew, Christopher E. Overton
https://arxiv.org/abs/2506.15528

Understanding the rift between update rules in Evolutionary Graph Theory: The intrinsic death rate drives star graphs from amplifying to suppressing natural selection
Evolutionary graph theory is the study of evolutionary dynamics in structured populations. A well-known problem in evolutionary graph theory is that the spread of mutation (measured by fixation probability) is impacted by the graph type chosen and the update rule. For example, the star graph is an amplifier of natural selection under the birth-death with fitness on birth (Bd) update rule but a suppressor of natural selection under the death-birth with fitness on birth (dB) update rule. A contin…

A theory of vectorial probability for quantum correlations
Sheng Feng
https://arxiv.org/abs/2506.12101 https://arxiv.org/pdf/2506.121…

A theory of vectorial probability for quantum correlations
Randomness is a ubiquitous phenomenon that is practically accompanied by physical events described by probability theory. However, probability by definition in the theory is a nonnegative scalar quantity. Here, we propose the concept of vectorial probability, quantified as a vector with interesting but hidden geometry, and develop a theory to describe random events with correlations dictated by this geometry. Based on this theory, we construct a local model that is able to reproduce the predict…

Non-exchangeable mean-field theory for adaptive weights: propagation of chaos and graphon sampling lemma
Datong Zhou
https://arxiv.org/abs/2506.13587 https…

Non-exchangeable mean-field theory for adaptive weights: propagation of chaos and graphon sampling lemma
We develop a mean-field theory for large, non-exchangeable particle (agent) systems where the states and interaction weights co-evolve in a coupled system of SDEs. A first main result is a generalization of the propagation of chaos. The weight adaptation in the SDEs makes the classical approach of using a static probability space as the continuum limit for the agent labels inadequate. We address this by introducing a label space endowed with a filtration that captures the stochasticity. While t…

This https://arxiv.org/abs/2412.19295 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Random matrix statistics and zeroes of $L$-functions via probability in $λ$-rings
We introduce a theory of probability in $λ$-rings designed to efficiently describe random variables valued in multisets of complex numbers, varieties over a field, or other similar enriched settings. A key role is played by the $σ$-moment generating function based on the plethystic exponential, which allows us to describe distributions and argue with independence in a way that is as simple as classical probability theory. As a first application, we use this theory to obtain a concise descript…

On the dynamical evolution of randomness Part B: Geometrisation and the origin of convergence in LLN
Allen Lobo
https://arxiv.org/abs/2506.14804 https://…

On the dynamical evolution of randomness Part B: Geometrisation and the origin of convergence in LLN
In classical probability theory, the convergence of empirical frequencies to theoretical probabilities: as captured by the Law of Large Numbers (LLN): is treated as axiomatic and emergent from statistical assumptions such as independence and identical distribution. In this work, a novel dynamical framework is constructed in which convergence arises as a consequence of structured evolution in outcome space, rather than a statistical postulate. Through this formalism, statistical convergence is d…

Wasserstein Distances on Quantum Structures: an Overview
Emily Beatty
https://arxiv.org/abs/2506.09794 https://arxiv.org/pdf/2506.097…

Wasserstein Distances on Quantum Structures: an Overview
The theory of optimal transport of probability measures has wide-ranging applications across a number of different fields, including concentration of measure, machine learning, Markov chains, and economics. The generalisation of optimal transport tools from probability measures to quantum states has shown great promise over the last few years, particularly in the development of the theory of Wasserstein-style distances and divergences between quantum states. Such distances have already led to a…

Weak* decomposition and Radon-Nikodym theorem for quantum expectations
Fouad Naderi
https://arxiv.org/abs/2506.12018 https://arxiv.or…

Weak* decomposition and Radon-Nikodym theorem for quantum expectations
A quantum expectation is a positive linear functional of norm one on a non-commutative probability space (i.e., a C*-algebra). For a given pair of quantum expectations $μ$ and $λ$ on a non-commutative probability space $A$, we propose a definition for weak* continuity and weak* singularity of $μ$ with respect to $λ$. Then, using the theory of von Neumann algebras, we obtain the natural weak* continuous and weak* singular parts of $μ$ with respect to $λ$. If $λ$ satisfies a weak tracial p…

A Unified Theory of Compositionality, Modularity, and Interpretability in Markov Decision Processes
Thomas J. Ringstrom, Paul R. Schrater
https://arxiv.org/abs/2506.09499

A Unified Theory of Compositionality, Modularity, and Interpretability in Markov Decision Processes
We introduce Option Kernel Bellman Equations (OKBEs) for a new reward-free Markov Decision Process. Rather than a value function, OKBEs directly construct and optimize a predictive map called a state-time option kernel (STOK) to maximize the probability of completing a goal while avoiding constraint violations. STOKs are compositional, modular, and interpretable initiation-to-termination transition kernels for policies in the Options Framework of Reinforcement Learning. This means: 1) STOKs can…

This https://arxiv.org/abs/2502.10523 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_qu…

The Quantum Wave Function as a Complex Probability Distribution
This paper suggests an interpretation for the wave function, based on some elements of Nelson's stochastic mechanics, the time-symmetric laws of quantum theory and complex probabilities. Our main purpose is to demonstrate that the wave function and its complex conjugate can be interpreted as complex probability densities (or quasi-probability distributions) related to non-real forward and backward in time stochastic motion respectively. Those two quasi-Brownian motions serve as mathematical abs…

FERRET: Private Deep Learning Faster And Better Than DPSGD
David Zagardo
https://arxiv.org/abs/2506.05416 https://arxiv.org/pdf/2506.…

FERRET: Private Deep Learning Faster And Better Than DPSGD
We revisit 1-bit gradient compression through the lens of mutual-information differential privacy (MI-DP). Building on signSGD, we propose FERRET--Fast and Effective Restricted Release for Ethical Training--which transmits at most one sign bit per parameter group with Bernoulli masking. Theory: We prove each fired group leaks at most ln 2 nats; after subsampling with rate s, the total privacy loss of G groups trained for T steps with firing probability p is epsilon = G * T * s * p * ln 2. Thu…

This https://arxiv.org/abs/2505.10981 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csAI_…

Rethinking the Role of Prompting Strategies in LLM Test-Time Scaling: A Perspective of Probability Theory
Recently, scaling test-time compute on Large Language Models (LLM) has garnered wide attention. However, there has been limited investigation of how various reasoning prompting strategies perform as scaling. In this paper, we focus on a standard and realistic scaling setting: majority voting. We systematically conduct experiments on 6 LLMs $\times$ 8 prompting strategies $\times$ 6 benchmarks. Experiment results consistently show that as the sampling time and computational overhead increase, co…

Asymptotic growth of the number of Reciprocal Classes in the Hecke Groups
Debattam Das, Krishnendu Gongopadhyay
https://arxiv.org/abs/2506.08449 https://…

Asymptotic growth of the number of Reciprocal Classes in the Hecke Groups
We estimate the asymptotic growth of reciprocal conjugacy classes in Hecke groups using their free product structure and word lengths of reciprocal elements. Our approach is different from other works in this direction and uses tools from basic probability theory.

A theory for diffusion-controlled reactions within nonequilibrium steady-states
Seokjin Moon, David T. Limmer
https://arxiv.org/abs/2505.22623 https://

A theory for diffusion-controlled reactions within nonequilibrium steady-states
We study diffusion-controlled processes in nonequilibrium steady-states, where standard rate theory assumptions break down. Using transition path theory, we generalize the relations between reactive probability fluxes and measures of the rate of the reaction. Stochastic thermodynamics analysis reveals how work constrains the enhancement of rates relative to their equilibrium values. An analytically solvable ion-pairing model under a strong electric field illustrates and validates our approach a…

This https://arxiv.org/abs/2409.00532 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Bounds on $T_c$ in the Eliashberg theory of Superconductivity. II: Dispersive phonons
The standard Eliashberg theory of superconductivity is studied, in which the effective electron-electron interactions are mediated by generally dispersive phonons, with Eliashberg spectral function $α^2 F(ω)\geq 0$ that is $\proptoω^2$ for small $ω>0$ and vanishes for large $ω$. The Eliashberg function also defines the electron-phonon coupling strength $λ:= 2 \int_0^\infty\frac{α^2 F(ω)}ωdω$. Setting $\frac{2α^2 F(ω)}ωdω=: λP(dω)$, formally defining a probability measure $P(dω)…

This https://arxiv.org/abs/2208.14711 has been replaced.
link: https://scholar.google.com/scholar?q=a

On real roots of polynomial systems of equations in the context of group theory
The probability that a zero of a random real polynomial of increasing degree is real tends to zero. However, passing from polynomials to Laurent polynomials yields a surprising result: the probability that a root is real tends not to zero, but to $1/\sqrt{3}$. A similar phenomenon has also been observed for systems of Laurent polynomials in several variables. By considering Laurent polynomials as functions associated with torus representations, we describe an analogous phenomenon for representa…

Temperature-Noise Interplay in a Coupled Model of Opinion Dynamics
Anna Chmiel, Julian Sienkiewicz
https://arxiv.org/abs/2506.07680 https://

Temperature-Noise Interplay in a Coupled Model of Opinion Dynamics
We consider a coupled system mimicking opinion formation under the influence of a group of $q$ neighbors ($q$-lobby) that consists of an Ising part governed by temperature-like parameter $T$ and a voter dynamics parameterized by noise probability $p$ (independence of choice). Using rigorous analytical calculations backed by extensive Monte Carlo simulations, we examine the interplay between these two quantities. Based on the theory of phase transitions, we derive the relation between $T$ and $p…

This https://arxiv.org/abs/2412.13751 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Notions of entropy for unitary representations
Several notions of entropy are studied widely in the ergodic theory of probability-preserving actions of countable groups. These include the generalization of Kolmogorov--Sinai entropy for amenable groups using Følner sequences, percolative entropy entropy for general countable groups, and Bowen's sofic entropy for sofic groups. In this work we pursue these notions across the analogy between ergodic theory and representation theory. We arrive at new quantities associated to unitary represent…

Understanding the Monty Hall Problem Through a Quantum Measurement Analogy
Mo Li
https://arxiv.org/abs/2506.00012 https://arxiv.org/p…

Understanding the Monty Hall Problem Through a Quantum Measurement Analogy
The Monty Hall problem is a classic probability puzzle known for its counterintuitive solution, revealing fundamental discrepancies between mathematical reasoning and human intuition. To bridge this gap, we introduce a novel explanatory framework inspired by quantum measurement theory. Specifically, we conceptualize the hosts' actions-opening doors to reveal non-prizes-as analogous to quantum measurements that cause asymmetric collapses of the probability distribution. This quantum-inspired int…

Existence, equivalence and spectrality of infinite convolutions in $\R^d$
Junjie Miao, Hongbo Zhao
https://arxiv.org/abs/2506.06670 https://

Existence, equivalence and spectrality of infinite convolutions in $\R^d$
In this paper, we study existence, equivalence and spectrality of infinite convolutions which may not be compactly supported in $d$-dimensional Euclidean space by manipulating various techniques in probability theory. First, we define the equivalent sequences, and we prove that the infinite convolutions converges simultaneously if they are generated by equivalent sequences. Moreover, the equi-positivity keeps unchanged for infinite convolutions generated by equivalent sequences. Next, we study …

Muscle Crossbridge Theory With Internal Crossbridge Dynamics
Mengjian Hua, Charles S. Peskin
https://arxiv.org/abs/2505.20198 https://

Muscle Crossbridge Theory With Internal Crossbridge Dynamics
We describe in this paper a crossbridge model in which an attached crossbridge behaves like a linear spring with a variable rest length. We assume in particular that the rest length has a linear force-velocity relation, and that the force and rest length are both zero at the moment of crossbridge attachment. Crossbridges that are not attached in our model have a fixed probability per unit time of attachment, and attached crossbridges have a probability per unit time of detachment that is a func…

Simplicial methods in the resource theory of contextuality
Aziz Kharoof, Cihan Okay
https://arxiv.org/abs/2505.24010 https://arxiv.or…

Simplicial methods in the resource theory of contextuality
We develop a resource theory of contextuality within the framework of symmetric monoidal categories, extending recent simplicial approaches to quantum contextuality. Building on the theory of simplicial distributions, which integrates homotopy-theoretic structures with probability, we introduce event scenarios as a functorial generalization of presheaf-theoretic measurement scenarios and prove their equivalence to bundle scenarios via the Grothendieck construction. We define symmetric monoidal …

This https://arxiv.org/abs/2502.16492 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Convergence of Clipped SGD on Convex $(L_0,L_1)$-Smooth Functions
We study stochastic gradient descent (SGD) with gradient clipping on convex functions under a generalized smoothness assumption called $(L_0,L_1)$-smoothness. Using gradient clipping, we establish a high probability convergence rate that matches the SGD rate in the $L$ smooth case up to polylogarithmic factors and additive terms. We also propose a variation of adaptive SGD with gradient clipping, which achieves the same guarantee. We perform empirical experiments to examine our theory and algor…

This https://arxiv.org/abs/2410.13738 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_sta…

Instance-dependent Convergence Theory for Diffusion Models
Score-based diffusion models have demonstrated outstanding empirical performance in machine learning and artificial intelligence, particularly in generating high-quality new samples from complex probability distributions. Improving the theoretical understanding of diffusion models, with a particular focus on the convergence analysis, has attracted significant attention. In this work, we develop a convergence rate that is adaptive to the smoothness of different target distributions, referred to …

Generalized Labeled Multi-Bernoulli Filters and Multitarget-Correlation Models
Ronald Mahler
https://arxiv.org/abs/2506.02772 https://

Generalized Labeled Multi-Bernoulli Filters and Multitarget-Correlation Models
The generalized labeled multi-Bernoulli (GLMB) filter is a theoretically rigorous Bayes-optimal multitarget tracking algorithm with computationally tractable implementations, based on labeled random finite set (LRFS) theory. It presumes that multitarget populations can be approximated using GLMB multitarget probability density functions (p.d.f.'s), which consist of weighted hypotheses regarding the current target-states. A special case of the GLMB p.d.f.-the LMB p.d.f.-presumes that the targets…

This https://arxiv.org/abs/2505.22623 has been replaced.
initial toot: https://mastoxiv.page/@ar…

A theory for diffusion-controlled reactions within nonequilibrium steady-states
We study diffusion-controlled processes in nonequilibrium steady-states, where standard rate theory assumptions break down. Using transition path theory, we generalize the relations between reactive probability fluxes and measures of the rate of the reaction. Stochastic thermodynamics analysis reveals how work constrains the enhancement of rates relative to their equilibrium values. An analytically solvable ion-pairing model under a strong electric field illustrates and validates our approach a…

Critical points of random polynomials and finite free cumulants
Octavio Arizmendi, Andrew Campbell, Katsunori Fujie
https://arxiv.org/abs/2506.08910 https:…

Critical points of random polynomials and finite free cumulants
A result of Hoskins and Steinerberger [Int. Math. Res. Not., (13):9784-9809, 2022] states that repeatedly differentiating a random polynomials with independent and identically distributed mean zero and variance one roots will result, after an appropriate rescaling, in a Hermite polynomial. We use the theory of finite free probability to extend this result in two natural directions: (1) We prove central limit theorems for the fluctuations around these deterministic limits for the polynomials and…

Simulating discrete-time quantum walk with urn model
Surajit Saha
https://arxiv.org/abs/2506.06896 https://arxiv.org/pdf/2506.06896…

Simulating discrete-time quantum walk with urn model
Urn models have long been used to study stochastic processes, probability distributions, and reinforcement dynamics. Meanwhile, discrete time quantum walks (DTQW) serve as fundamental components in quantum computation and quantum information theory. This work explores a novel connection between an urn model and discrete-time quantum walks, focusing on how urn-based stochastic processes can provide insights into quantum state evolution and algorithmic behavior. Importantly, our model operates en…

Hydrodynamic fluctuations of stochastic charged cellular automata
Takato Yoshimura, \v{Z}iga Krajnik
https://arxiv.org/abs/2506.05247 https://

Hydrodynamic fluctuations of stochastic charged cellular automata
We study charge fluctuations of a family of stochastic charged cellular automata away from the deterministic single-file limit and obtain the exact typical charge probability distributions, known to be anomalous, using hydrodynamics. The cellular automata considered are examples of linearly degenerate systems where two distinct mechanisms of diffusion, namely normal and convective diffusion, coexist. Our formalism, based on macroscopic fluctuation theory, allows us to describe current fluctuati…

Fr\"ohlich Condensation of Bosons: Graph texture of curl flux network for nonequilibrium properties
Feihong Liu, Chase Slowey, Xuanhua Wang, Dangyuan Lei, Zhiyue Lu, Zhedong Zhang
https://arxiv.org/abs/2506.00792

Fröhlich Condensation of Bosons: Graph texture of curl flux network for nonequilibrium properties
Nonequilibrium condensates of bosons subject to energy pump and dissipation are investigated, manifesting the Fröhlich coherence proposed in 1968. A quantum theory is developed to capture such a nonequilibrium nature, yielding a certain graphic structure arising from the detailed-balance breaking. The results show a network of probability curl fluxes that reveals a graph topology. The winding number associated with the flux network is thus identified as a new order parameter for the phase tran…

Percolation on random 2-lifts
Paul Drouvill\'e
https://arxiv.org/abs/2506.01612 https://arxiv.org/pdf/2506.01612

Percolation on random 2-lifts
Given a graph $G$, we consider a model for a random cover of $G$ by taking two parallel copies of $G$ and crossing every pair of parallel edges randomly with probability $q$ independently of each other. The resulting graph $G_q$, is a random $2$-lift of $G$ that may not be transitive but still probabilistically exhibit many properties of transitive graphs. Studying percolation in this context can help us test the reliability and robustness of our proofs methods in percolation theory. Our three …