Journal of Statistical Physics

Kovchegov, Yevgeniy; Otto, Peter T.

doi: 10.1007/s10955-023-03188-2pmid: N/A

In this paper we show the existence of the minimal solution to the multidimensional Lambert–Euler inversion, a multidimensional generalization of [-e-1,0)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$[-e^{-1},0)$$\end{document} branch of Lambert W function W0(x)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$W_0(x)$$\end{document}. Specifically, for a given nonnegative irreducible symmetric matrix V∈Rk×k\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$V \in \mathbb {R}^{k \times k}$$\end{document} and a vector u∈(0,∞)k\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textbf{u}\in (0,\infty )^k$$\end{document}, we show that, if the system of equations yjexp{-ejTVy}=uj∀j=1,…,k,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} y_j \exp \big \{-\textbf{e}_j^{\textsf {T}} V \textbf{y} \big \} = u_j \qquad \forall j=1,\ldots ,k, \end{aligned}$$\end{document}has at least one solution, it must have a minimal solution y∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textbf{y}^*$$\end{document}, where the minimum is achieved in all coordinates yj\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$y_j$$\end{document} simultaneously. Moreover, such y∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textbf{y}^*$$\end{document} is the unique solution satisfying ρVD[yj∗]≤1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\rho \left( V D[y^*_j] \right) \le 1$$\end{document}, where D[yj∗]=diag(yj∗)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$D[y^*_j]=\textsf {diag}(y_j^*)$$\end{document} is the diagonal matrix with entries yj∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$y^*_j$$\end{document} and ρ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\rho $$\end{document} denotes the spectral radius. Our main application is in the analysis of the vector-multiplicative coalescent process. It is a coalescent process with k types of particles and k-dimensional vector-valued cluster weights representing the composition of a cluster by particle types. The clusters merge according to the vector-multiplicative kernel K(x,y)=xTVy\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$K(\textbf{x}, \textbf{y})=\textbf{x}^{\textsf {T}} V \textbf{y}$$\end{document}. First, we derive some new combinatorial results, and use them to solve the corresponding modified Smoluchowski equations obtained as a hydrodynamic limit of vector-multiplicative coalescent. Then, we use multidimensional Lambert–Euler inversion to establish gelation and find a closed form expression for the gelation time. We also find the asymptotic length of the minimal spanning tree for a broad range of graphs equipped with random edge lengths.

Cai, Yifan; Wang, Tao; Zhao, Liu

doi: 10.1007/s10955-023-03204-5pmid: N/A

Two different versions of relativistic Langevin equation in curved spacetime background are constructed, both are manifestly general covariant. It is argued that, from the observer’s point of view, the version which takes the proper time of the Brownian particle as evolution parameter contains some conceptual issues, while the one which makes use of the proper time of the observer is more physically sound. The two versions of the relativistic Langevin equation are connected by a reparametrization scheme. In spite of the issues contained in the first version of the relativistic Langevin equation, it still permits to extract the physical probability distributions of the Brownian particles, as is shown by Monte Carlo simulation in the example case of Brownian motion in (1+1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(1+1)$$\end{document}-dimensional Minkowski spacetime.

Konno, Norio; Matsue, Kaname; Segawa, Etsuo

doi: 10.1007/s10955-023-03211-6pmid: N/A

We propose an intermediate walk continuously connecting an open quantum random walk and a quantum walk with parameters M∈N\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$M\in {{\mathbb {N}}}$$\end{document} controlling a decoherence effect; if M=1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$M=1$$\end{document}, the walk coincides with an open quantum random walk, while M=∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$M=\infty $$\end{document}, the walk coincides with a quantum walk. We define a measure which recovers usual probability measures on Z\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{\mathbb {Z}}}$$\end{document} for M=∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$M=\infty $$\end{document} and M=1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$M=1$$\end{document} and we observe intermediate behavior through numerical simulations for varied positive values M. In the case for M=2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$M=2$$\end{document}, we analytically show that a typical behavior of quantum walks appears even in a small gap of the parameter from the open quantum random walk. More precisely, we observe both the ballistically moving towards left and right sides and localization of this walker simultaneously. The analysis is based on Kato’s perturbation theory for linear operator. We further analyze this limit theorem in more detail and show that the above three modes are described by Gaussian distributions.

Zhou, Jianwen; Zhou, Hai-Jun

doi: 10.1007/s10955-023-03210-7pmid: N/A

The K-core of a graph is the unique maximum subgraph within which each vertex connects to K or more other vertices. The optimal K-core attack problem asks to delete the minimum number of vertices from the K-core to induce its complete collapse. A hierarchical cycle-tree packing model is introduced here for this challenging combinatorial optimization problem. We convert the temporally long-range correlated K-core pruning dynamics into locally tree-like static patterns and analyze this model through the replica-symmetric cavity method of statistical physics. A set of coarse-grained belief propagation equations are derived to predict single vertex marginal probabilities efficiently. The associated hierarchical cycle-tree guided attack (hCTGA) algorithm is able to construct nearly optimal attack solutions for regular random graphs and Erdös-Rényi random graphs. Our cycle-tree packing model may also be helpful for constructing optimal initial conditions for other irreversible dynamical processes on sparse random graphs.

Chen, Qi; Wang, Hong-tao; Guo, Chao

doi: 10.1007/s10955-023-03209-0pmid: N/A

Hamiltonian approach in quantum mechanics provides a new thinking for barrier option pricing. For proportional floating barrier step options, the option price changing process is similar to the one dimensional trapezoid potential barrier scattering problem in quantum mechanics; for floating double-barrier step options, the option price changing process is analogous to a particle moving in a finite symmetric square potential well. Using Hamiltonian methodology, the analytical expressions of pricing kernel and option price could be derived. Numerical results of option price as a function of underlying price, floating rate, interest rate and exercise price are shown, which are consistent with the results given by mathematical calculations.

Wang, Hao

doi: 10.1007/s10955-023-03208-1pmid: N/A

We study the Vlasov–Poisson–Boltzmann system without angular cutoff and the Vlasov–Poisson–Landau system with all hard potentials in the perturbation setting, and establish the Gevrey smoothness in both spatial and velocity variables for a class of low-regularity weak solutions. This work extends the results by Duan–Li–Liu [16] for the pure Boltzmann equation to the case of the Vlasov–Poisson–Boltzmann/Landau systems.

Di Crescenzo, Antonio; Iuliano, Antonella; Mustaro, Verdiana; Verasani, Gabriella

doi: 10.1007/s10955-023-03189-1pmid: N/A

We investigate the effects of the resetting mechanism to the origin for a random motion on the real line characterized by two alternating velocities v1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$v_1$$\end{document} and v2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$v_2$$\end{document}. We assume that the sequences of random times concerning the motions along each velocity follow two independent geometric counting processes of intensity λ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\lambda $$\end{document}, and that the resetting times are Poissonian with rate ξ>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\xi >0$$\end{document}. Under these assumptions we obtain the probability laws of the modified telegraph process describing the position and the velocity of the running particle. Our approach is based on the Markov property of the resetting times and on the knowledge of the distribution of the intertimes between consecutive velocity changes. We obtain also the asymptotic distribution of the particle position when (i) λ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\lambda $$\end{document} tends to infinity, and (ii) the time goes to infinity. In the latter case the asymptotic distribution arises properly as an effect of the resetting mechanism. A quite different behavior is observed in the two cases when v2<0<v1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$v_2<0<v_1$$\end{document} and 0<v2<v1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$0<v_2<v_1$$\end{document}. Furthermore, we focus on the determination of the moment-generating function and on the main moments of the process describing the particle position under reset. Finally, we analyse the mean-square distance between the process subject to resets and the same process in absence of resets. Quite surprisingly, the lowest mean-square distance can be found for ξ=0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\xi =0$$\end{document}, for a positive ξ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\xi $$\end{document}, or for ξ→+∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\xi \rightarrow +\infty $$\end{document} depending on the choice of the other parameters.

Parvaneh, Azadeh; Roy, Rahul

doi: 10.1007/s10955-023-03213-4pmid: N/A

We study two directed non-planar random graphs, each of which has a dependence structure. We prove that each of these models, under diffusive scaling, converges to the Brownian web. To obtain this, we first obtain a Markovian renewal structure of the paths of the graph and then study the coalescence time of any two paths. Finally, we show that the condition required by Coletti and Valle (Ann Inst Henri Poincaré Prob Stat 50:899–919, 2014) in their study of the diffusive scaling limit of a generalized Howard model of drainage can be relaxed.

Friedrich, Manuel; Kreutz, Leonard

doi: 10.1007/s10955-023-03202-7pmid: N/A

We devise a new technique to prove two-dimensional crystallization results in the square lattice for finite particle systems. We apply this strategy to energy minimizers of configurational energies featuring two-body short-ranged particle interactions and three-body angular potentials favoring bond-angles of the square lattice. To each configuration, we associate its bond graph which is then suitably modified by identifying chains of successive atoms. This method, called stratification, reduces the crystallization problem to a simple minimization that corresponds to a proof via slicing of the isoperimetric inequality in ℓ1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\ell ^1$$\end{document}. As a byproduct, we also prove a fluctuation estimate for minimizers of the configurational energy, known as the n3/4\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n^{3/4}$$\end{document}-law.

Fukushima-Kimura, Bruno Hideki; Kawamoto, Noe; Noda, Eitaro; Sakai, Akira

doi: 10.1007/s10955-023-03179-3pmid: N/A

The Digital Annealer is a CMOS hardware designed by Fujitsu Laboratories for high-speed solving of Quadratic Unconstrained Binary Optimization (QUBO) problems that could be difficult to solve by means of existing general-purpose computers. In this paper, we present a mathematical description of the first-generation Digital Annealer’s Algorithm from the Markov chain theory perspective, establish a relationship between its stationary distribution with the Gibbs-Boltzmann distribution, and provide a necessary and sufficient condition on its cooling schedule that ensures asymptotic convergence to the ground states.