Probability

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New submissions for Thu, 9 May 24

[1]  arXiv:2405.04835 [pdf, ps, other]

Title: Precise large deviation for stationary sequence of branching process with immigration

Authors: Jiayan Guo, Wenming Hong

Comments: arXiv admin note: substantial text overlap with arXiv:2405.04073

Subjects: Probability (math.PR)

It is known that there exists the stationary sequence of branching process with immigration $\{X_{n}\}_{n\in\mathbb{Z}}$ under some conditions (Foster and Williamson (1971)), when the offspring is critical or subcritical. A precise large deviation probability for the partial sum $S_{n}=X_{1}+\cdots+X_{n}$ is specified, the significant difference is revealed for the critical and subcritical cases.

[2]  arXiv:2405.04951 [pdf, other]

Title: Gaussian consensus processes and their Lyapunov exponents

Authors: Edward Crane, Stanislav Volkov

Subjects: Probability (math.PR)

We introduce a simple dynamic model of opinion formation, in which a finite population of individuals hold vector-valued opinions. At each time step, each individual's opinion moves towards the mean opinion but is then perturbed independently by a centred multivariate Gaussian random variable, with covariance proportional to the covariance matrix of the opinions of the population. We establish precise necessary and sufficient conditions on the parameters of the model, under which all opinions converge to a common limiting value. Asymptotically perfect correlation emerges between opinions on different topics. Our results are rigorous and based on properties of the partial products of an i.i.d. sequence of random matrices. Each matrix is a fixed linear combination of the identity matrix and a real Ginibre matrix. We derive an analytic expression for the maximal Lyapunov exponent of this product sequence. We also analyze a continuous-time analogue of our model.

[3]  arXiv:2405.04999 [pdf, ps, other]

Title: Small ball probability for multiple singular values of symmetric random matrices

Authors: Yi Han

Subjects: Probability (math.PR)

Let $A_n$ be an $n\times n$ random symmetric matrix with $(A_{ij})_{i< j}$ i.i.d. mean $0$, variance 1, following a subGaussian distribution and diagonal elements i.i.d. following a subGaussian distribution with a fixed variance. We investigate the joint small ball probability that $A_n$ has eigenvalues near two fixed locations $\lambda_1$ and $\lambda_2$, where $\lambda_1$ and $\lambda_2$ are sufficiently separated and in the bulk of the semicircle law. More precisely we prove that for a wide class of entry distributions of $A_{ij}$ that involve all Gaussian convolutions (where $\sigma_{min}(\cdot)$ denotes the least singular value of a square matrix), $$\mathbb{P}(\sigma_{min}(A_n-\lambda_1 I_n)\leq\delta_1n^{-1/2},\sigma_{min}(A_n-\lambda_2 I_n)\leq\delta_2n^{-1/2})\leq c\delta_1\delta_2+e^{-cn}.$$ The given estimate approximately factorizes as the product of the estimates for the two individual events, which is an indication of quantitative independence. The estimate readily generalizes to $d$ distinct locations. As an application, we upper bound the probability that there exist $d$ eigenvalues of $A_n$ asymptotically satisfying any fixed linear equation, which in particular gives a lower bound of the distance to this linear relation from any possible eigenvalue pair that holds with probability $1-o(1)$, and rules out the existence of two equal singular values in generic regions of the spectrum.

[4]  arXiv:2405.05045 [pdf, ps, other]

Title: Maximum of the Characteristic Polynomial of I.I.D. Matrices

Authors: Giorgio Cipolloni, Benjamin Landon

Comments: 85 pages

Subjects: Probability (math.PR); Mathematical Physics (math-ph)

We compute the leading order asymptotic of the maximum of the characteristic polynomial for i.i.d. matrices with real or complex entries. In particular, this result is new even for real Ginibre matrices, which was left as an open problem in [arXiv:2303.09912]; the complex Ginibre case was covered in [arXiv:1902.01983]. These are the first universality results for the non--Hermitian analog of the first order term of the Fyodorov--Hiary--Keating conjecture. Our methods are based on constructing a coupling to the branching random walk via Dyson Brownian motion. In particular, we find a new connection between real i.i.d. matrices and inhomogeneous branching random walk.

[5]  arXiv:2405.05102 [pdf, ps, other]

Title: The Harmonic Descent Chain

Authors: David J. Aldous, Svante Janson, Xiaodan Li

Comments: 14 pages

Subjects: Probability (math.PR)

The decreasing Markov chain on \{1,2,3, \ldots\} with transition probabilities $p(j,j-i) \propto 1/i$ arises as a key component of the analysis of the beta-splitting random tree model. We give a direct and almost self-contained "probability" treatment of its occupation probabilities, as a counterpart to a more sophisticated but perhaps opaque derivation using a limit continuum tree structure and Mellin transforms.

[6]  arXiv:2405.05203 [pdf, ps, other]

Title: A multiple coupon collection process and its Markov embedding structure

Authors: Ellen Baake (Bielefeld), Michael Baake (Bielefeld)

Comments: 19 pages

Subjects: Probability (math.PR)

The embedding problem of Markov transition matrices into Markov semigroups is a classic problem that regained a lot of impetus and activities in recent years. We consider it here for the following generalisation of the well-known coupon collection process: from a finite set of distinct objects, a subset is drawn repeatedly according to some probability distribution, independently and with replacement, and each time united with the set of objects sampled so far. We derive and interpret properties and explicit conditions for the resulting discrete-time Markov chain to be representable within a semigroup or a flow of a continuous-time process of the same type.

[7]  arXiv:2405.05246 [pdf, ps, other]

Title: Semi-infinite particle systems with exclusion interaction and heterogeneous jump rates

Authors: Mikhail Menshikov, Serguei Popov, Andrew Wade

Comments: 32 pages, 1 figure

Subjects: Probability (math.PR)

We study semi-infinite particle systems on the one-dimensional integer lattice, where each particle performs a continuous-time nearest-neighbour random walk, with jump rates intrinsic to each particle, subject to an exclusion interaction which suppresses jumps that would lead to more than one particle occupying any site. Under appropriate hypotheses on the jump rates (uniformly bounded rates is sufficient) and started from an initial condition that is a finite perturbation of the close-packed configuration, we give conditions under which the particles evolve as a single, semi-infinite "stable cloud". More precisely, we show that inter-particle separations converge to a product-geometric stationary distribution, and that the location of every particle obeys a strong law of large numbers with the same characteristic speed.

Cross-lists for Thu, 9 May 24

[8]  arXiv:2405.04670 (cross-list from math.CO) [pdf, other]

Title: Isomorphisms between random $d$-hypergraphs

Authors: Théo Lenoir

Comments: 13 pages, 1 figure

Subjects: Combinatorics (math.CO); Probability (math.PR)

We characterize the size of the largest common induced subgraph of two independent random uniform $d$-hypergraphs of different sizes with $d\geq 3$. More precisely, its distribution is asymptotically concentrated on two points, and we obtain as a consequence a phase transition for the inclusion of the smallest hypergraph in the largest one. This generalizes to uniform random $d$-hypergraphs the results of Chatterjee and Diaconis for uniform random graphs.
Our proofs rely on the first and second moment methods.

[9]  arXiv:2405.05082 (cross-list from math.CO) [pdf, ps, other]

Title: On linear-combinatorial problems associated with subspaces spanned by $\{\pm 1\}$-vectors

Authors: Anwar A. Irmatov

Comments: 13 pages

Subjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM); Algebraic Topology (math.AT); Probability (math.PR)

A complete answer to the question about subspaces generated by $\{\pm 1\}$-vectors, which arose in the work of I.Kanter and H.Sompolinsky on associative memories, is given. More precisely, let vectors $v_1, \ldots , v_p,$ $p\leq n-1,$ be chosen at random uniformly and independently from $\{\pm 1\}^n \subset {\bf R}^n.$ Then the probability ${\mathbb P}(p, n)$ that $$span \ \langle v_1, \ldots , v_p \rangle \cap \left\{ \{\pm 1\}^n \setminus \{\pm v_1, \ldots , \pm v_p\}\right\} \ne \emptyset \ $$ is shown to be $$4{p \choose 3}\left(\frac{3}{4}\right)^n + O\left(\left(\frac{5}{8} + o_n(1)\right)^n\right) \quad \mbox{as} \quad n\to \infty,$$ where the constant implied by the $O$-notation does not depend on $p$. The main term in this estimate is the probability that some 3 vectors $v_{j_1}, v_{j_2}, v_{j_3}$ of $v_j$, $j= 1, \ldots , p,$ have a linear combination that is a $\{\pm 1\}$-vector different from $\pm v_{j_1}, \pm v_{j_2}, \pm v_{j_3}. $

[10]  arXiv:2405.05192 (cross-list from math.NA) [pdf, other]

Title: Full error analysis of the random deep splitting method for nonlinear parabolic PDEs and PIDEs with infinite activity

Authors: Ariel Neufeld, Philipp Schmocker, Sizhou Wu

Subjects: Numerical Analysis (math.NA); Machine Learning (cs.LG); Probability (math.PR); Mathematical Finance (q-fin.MF)

In this paper, we present a randomized extension of the deep splitting algorithm introduced in [Beck, Becker, Cheridito, Jentzen, and Neufeld (2021)] using random neural networks suitable to approximately solve both high-dimensional nonlinear parabolic PDEs and PIDEs with jumps having (possibly) infinite activity. We provide a full error analysis of our so-called random deep splitting method. In particular, we prove that our random deep splitting method converges to the (unique viscosity) solution of the nonlinear PDE or PIDE under consideration. Moreover, we empirically analyze our random deep splitting method by considering several numerical examples including both nonlinear PDEs and nonlinear PIDEs relevant in the context of pricing of financial derivatives under default risk. In particular, we empirically demonstrate in all examples that our random deep splitting method can approximately solve nonlinear PDEs and PIDEs in 10'000 dimensions within seconds.

[11]  arXiv:2405.05223 (cross-list from math.AP) [pdf, ps, other]

Title: The Harnack inequality fails for nonlocal kinetic equations

Authors: Moritz Kassmann, Marvin Weidner

Subjects: Analysis of PDEs (math.AP); Probability (math.PR)

We prove that the Harnack inequality fails for nonlocal kinetic equations. Such equations arise as linearized models for the Boltzmann equation without cutoff and are of hypoelliptic type. We provide a counterexample for the simplest equation in this theory, the fractional Kolmogorov equation. Our result reflects a purely nonlocal phenomenon since the Harnack inequality holds true for local kinetic equations like the Kolmogorov equation.

Replacements for Thu, 9 May 24

[12]  arXiv:2111.10875 (replaced) [pdf, other]

Title: The number of real zeros of elliptic polynomials

Authors: Nhan D. V. Nguyen

Comments: 49 pages, 2 figures, 1 table, final version

Subjects: Probability (math.PR)

[13]  arXiv:2310.10960 (replaced) [pdf, other]

Title: The half-space log-Gamma polymer in the bound phase

Authors: Sayan Das, Weitao Zhu

Comments: 40 pages, 17 figures; Final version; To appear in Communications in Mathematical Physics

Subjects: Probability (math.PR)

[14]  arXiv:2311.05979 (replaced) [pdf, ps, other]

Title: The Distribution of Polynomials in Monotone Independent Elements

Authors: Marwa Banna, Pei-Lun Tseng

Comments: 18 pages

Subjects: Probability (math.PR)

[15]  arXiv:2311.10625 (replaced) [pdf, ps, other]

Title: Central limit theorems for Soft random simplicial complexes

Authors: Julián David Candela

Comments: 28 pages

Subjects: Probability (math.PR); Algebraic Topology (math.AT)

[16]  arXiv:2403.01658 (replaced) [pdf, ps, other]

Title: Noise sensitivity and stability on groups

Authors: Ryokichi Tanaka

Subjects: Probability (math.PR); Discrete Mathematics (cs.DM)

[17]  arXiv:2404.16021 (replaced) [pdf, other]

Title: Critical beta-splitting, via contraction

Authors: Brett Kolesnik

Comments: v2: added references #5 and #7, in response to a private communication with Oleksandr Iksanov (see acknowledgments)

Subjects: Probability (math.PR); Combinatorics (math.CO)

[18]  arXiv:2404.18918 (replaced) [pdf, ps, other]

Title: Strong solutions to McKean-Vlasov SDEs associated to a class of degenerate Fokker-Planck equations with coefficients of Nemytskii-type

Authors: Sebastian Grube

Comments: 26 pages

Subjects: Probability (math.PR); Analysis of PDEs (math.AP)

[19]  arXiv:2405.00363 (replaced) [pdf, other]

Title: Competing bootstrap processes on the random graph $G(n,p)$

Authors: Michele Garetto, Emilio Leonardi, Giovanni Luca Torrisi

Subjects: Probability (math.PR); Dynamical Systems (math.DS)

[20]  arXiv:2307.01344 (replaced) [pdf, ps, other]

Title: Equidistribution of high traces of random matrices over finite fields and cancellation in character sums of high conductor

Authors: Ofir Gorodetsky, Valeriya Kovaleva

Comments: 19 pages, accepted version. Includes 1-page appendix on character sums not included in published version

Subjects: Number Theory (math.NT); Probability (math.PR)

[21]  arXiv:2311.08872 (replaced) [pdf, other]

Title: Multilevel Monte Carlo methods for the Dean-Kawasaki equation from Fluctuating Hydrodynamics

Authors: Federico Cornalba, Julian Fischer

Comments: 28 pages, 6 figures

Subjects: Numerical Analysis (math.NA); Analysis of PDEs (math.AP); Probability (math.PR)

[22]  arXiv:2312.17565 (replaced) [pdf, other]

Title: Thermodynamics of the five-vertex model with scalar-product boundary conditions

Authors: Ivan N. Burenev, Andrei G. Pronko

Comments: 52 pages, 7 figures; v2: Sections 1, 4, and 5 extended, references added

Subjects: Mathematical Physics (math-ph); Statistical Mechanics (cond-mat.stat-mech); Probability (math.PR)

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