搜索结果：random

We analyze the asymptotic fluctuations of linear eigenvalue statistics of random centrosymmetric matrices with i.i.d. entries. We prove that for a complex analytic test function, the centered and normalized linear eigenvalue statistics of random centrosymmetric matrices converge to a normal distribution. We find the exact expression of the variance of the limiting normal distribution via combinatorial arguments. Moreover, we also argue that the limiting spectral distribution of properly scaled centrosymmetric matrices follows the circular law.

In this paper, we examine a specific type of random chains and propose an unified approach to studying the degree-based topological indices, including their extreme values. We derive explicit analytical expressions for the expected values and variances of these indices and we establish the asymptotic behavior of the indices. Specifically, we analyze the first Zagreb index, Sombor index, Harmonic index, Geometric-Arithmetic index, Inverse Sum Index, and the second Zagreb index for various general random chains, including random phenylene, random polyphenyl, random hexagonal, and linear chains.

When each vertex is assigned a set, the intersection graph generated by the sets is the graph in which two distinct vertices are joined by an edge if and only if their assigned sets have a nonempty intersection. An interval graph is an intersection graph generated by intervals in the real line. A chordal graph can be considered as an intersection graph generated by subtrees of a tree. In 1999, Karoński, Scheinerman and Singer-Cohen [Combin Probab Comput 8 (1999), 131--159] introduced a random intersection graph by taking randomly assigned sets. The random intersection graph $G(n,m;p)$ has $n$ vertices and sets assigned to the vertices are chosen to be i.i.d. random subsets of a fixed set $M$ of size $m$ where each element of $M$ belongs to each random subset with probability $p$, independently of all other elements in $M$. Fill, Scheinerman and Singer-Cohen [Random Struct Algorithms 16 (2000), 156--176] showed that the total variation distance between the random graph $G(n,m;p)$ and the Erdös-Rényi graph $G(n,\hat{p})$ tends to $0$ for any $0 \leq p=p(n) \leq 1$ if $m=n^α$, $α>6$, where $\hat{p}$ is chosen so that the expected numbers of edges in the two graphs are the same. In

The Ising $p$-spin glass and random $k$-SAT are two canonical examples of disordered systems that play a central role in understanding the link between geometric features of optimization landscapes and computational tractability. Both models exhibit hard regimes where all known polynomial-time algorithms fail and possess the multi Overlap Gap Property ($m$-OGP), an intricate geometrical property that rigorously rules out a broad class of algorithms exhibiting input stability. We establish that, in both models, the symmetric $m$-OGP undergoes a sharp phase transition, and we pinpoint its exact threshold. For the Ising $p$-spin glass, our results hold for all sufficiently large $p$; for the random $k$-SAT, they apply to all $k$ growing mildly with the number of Boolean variables. Notably, our findings yield qualitative insights into the power of OGP-based arguments. A particular consequence for the Ising $p$-spin glass is that the strength of the $m$-OGP in establishing algorithmic hardness grows without bound as $m$ increases. These are the first sharp threshold results for the $m$-OGP. Our analysis hinges on a judicious application of the second moment method, enhanced by concentra

We consider two random matrix ensembles which are relevant for describing critical spectral statistics in systems with multifractal eigenfunction statistics. One of them is the Gaussian non-invariant ensemble which eigenfunction statistics is multifractal, while the other is the invariant random matrix ensemble with a shallow, log-square confinement potential. We demonstrate a close correspondence between the spectral as well as eigenfuncton statistics of these random matrix ensembles and those of the random tight-binding Hamiltonian in the point of the Anderson localization transition in three dimensions. Finally we present a simple field theory in 1+1 dimensions which reproduces level statistics of both of these random matrix models and the classical Wigner-Dyson spectral statistics in the framework of the unified formalism of Luttinger liquid. We show that the (equal-time) density correlations in both random matrix models correspond to the finite-temperature density correlations of the Luttinger liquid. We show that spectral correlations in the invariant ensemble with log-square confinement correspond to a Luttinger liquid in the 1+1 curved space-time with the event horizon, sim

We consider random walks in dynamic random environments given by Markovian dynamics on $\mathbb{Z}^d$. We assume that the environment has a stationary distribution $μ$ and satisfies the Poincaré inequality w.r.t. $μ$. The random walk is a perturbation of another random walk (called "unperturbed"). We assume that also the environment viewed from the unperturbed random walk has stationary distribution $μ$. Both perturbed and unperturbed random walks can depend heavily on the environment and are not assumed to be finite-range. We derive a law of large numbers, an averaged invariance principle for the position of the walker and a series expansion for the asymptotic speed. We also provide a condition for non-degeneracy of the diffusion, and describe in some details equilibrium and convergence properties of the environment seen by the walker. All these results are based on a more general perturbative analysis of operators that we derive in the context of $L^2$-bounded perturbations of Markov processes by means of the so-called Dyson-Phillips expansion.

Curve samplers are sampling algorithms that proceed by viewing the domain as a vector space over a finite field, and randomly picking a low-degree curve in it as the sample. Curve samplers exhibit a nice property besides the sampling property: the restriction of low-degree polynomials over the domain to the sampled curve is still low-degree. This property is often used in combination with the sampling property and has found many applications, including PCP constructions, local decoding of codes, and algebraic PRG constructions. The randomness complexity of curve samplers is a crucial parameter for its applications. It is known that (non-explicit) curve samplers using $O(\log N+\log(1/δ))$ random bits exist, where $N$ is the domain size and $δ$ is the confidence error. The question of explicitly constructing randomness-efficient curve samplers was first raised in \cite{TU06} where they obtained curve samplers with near-optimal randomness complexity. We present an explicit construction of low-degree curve samplers with {\em optimal} randomness complexity (up to a constant factor), sampling curves of degree $\left(m\log_q(1/δ)\right)^{O(1)}$ in $\mathbb{F}_q^m$. Our construction is a

We study limit distributions of independent random matrices as well as limit joint distributions of their blocks under normalized partial traces composed with classical expectation. In particular, we are concerned with the ensemble of symmetric blocks of independent Hermitian random matrices which are asymptotically free, asymptotically free from diagonal deterministic matrices, and whose norms are uniformly bounded. This class contains symmetric blocks of unitarily invariant Hermitian random matrices whose asymptotic distributions are compactly supported probability measures on the real line. Our approach is based on the concept of matricial freeness which is a generalization of freeness in free probability. We show that the associated matricially free Gaussian operators provide a unified framework for studying the limit distributions of sums and products of independent rectangular random matrices, including non-Hermitian Gaussian matrices and matrices of Wishart type. This framework also leads to random matrix models for boolean, monotone and s-free independences.

Let $x \in S^{n-1}$ be a unit eigenvector of an $n \times n$ random matrix. This vector is delocalized if it is distributed roughly uniformly over the real or complex sphere. This intuitive notion can be quantified in various ways. In these lectures, we will concentrate on the no-gaps delocalization. This type of delocalization means that with high probability, any non-negligible subset of the support of $x$ carries a non-negligible mass. Proving the no-gaps delocalization requires establishing small ball probability bounds for the projections of random vector. Using Fourier transform, we will prove such bounds in a simpler case of a random vector having independent coordinates of a bounded density. This will allow us to derive the no-gaps delocalization for matrices with random entries having a bounded density. In the last section, we will discuss the applications of delocalization to the spectral properties of Erdős-Rényi random graphs.

We prove that in all regular robust expanders $G$ every edge is asymptotically equally likely contained in a uniformly chosen perfect matching $M$. We also show that given any fixed matching or spanning regular graph $N$ in $G$, the random variable $|M\cap E(N)|$ is approximately Poisson distributed. This in particular confirms a conjecture and a question due to Spiro and Surya, and complements results due to Kahn and Kim who proved that in a regular graph every vertex is asymptotically equally likely contained in a uniformly chosen matching. Our proofs rely on the switching method and the fact that simple random walks mix rapidly in robust expanders.

We study behavior in space and time of random walks in an i.i.d. random environment on Z^d, d>=3. It is assumed that the measure governing the environment is isotropic and concentrated on environments that are small perturbations of the fixed environment corresponding to simple random walk. We develop a revised and extended version of the paper of Bolthausen and Zeitouni (2007) on exit laws from large balls, which, as we hope, is easier to follow. Further, we study mean sojourn times in balls. This work is part of the author's PhD thesis under the supervision of Erwin Bolthausen. A generalization of the results on exit measures to certain anisotropic random walks in random environment is available at arXiv:1309.3169.

We study random walks on the giant component of Hyperbolic Random Graphs (HRGs), in the regime when the degree distribution obeys a power law with exponent in the range $(2,3)$. In particular, we first focus on the expected time for a random walk to hit a given vertex or visit, i.e. cover, all vertices. We show that, a.a.s. (with respect to the HRG), and up to multiplicative constants: the cover time is $n(\log n)^2$, the maximum hitting time is $n\log n$, and the average hitting time is $n$. We then determine the expected time to commute between two given vertices a.a.s., up to a small factor polylogarithmic in $n$, and under some mild hypothesis on the pair of vertices involved. Our results are proved by controlling effective resistances using the energy dissipated by carefully designed network flows associated to a tiling of the hyperbolic plane, on which we overlay a forest-like structure.

We study the notion of local treewidth in sparse random graphs: the maximum treewidth over all $k$-vertex subgraphs of an $n$-vertex graph. When $k$ is not too large, we give nearly tight bounds for this local treewidth parameter; we also derive tight bounds for the local treewidth of noisy trees, trees where every non-edge is added independently with small probability. We apply our upper bounds on the local treewidth to obtain fixed parameter tractable algorithms (on random graphs and noisy trees) for edge-removal problems centered around containing a contagious process evolving over a network. In these problems, our main parameter of study is $k$, the number of initially ``infected'' vertices in the network. For the random graph models we consider and a certain range of parameters the running time of our algorithms on $n$-vertex graphs is $2^{o(k)}\textrm{poly}(n)$, improving upon the $2^{Ω(k)}\textrm{poly}(n)$ performance of the best-known algorithms designed for worst-case instances of these edge deletion problems.

The upper tail problem in a random graph asks to estimate the probability that the number of copies of some fixed subgraph in an Erdős--Rényi random graph exceeds its expectation by some constant factor. There has been much exciting recent progress on this problem. We study the corresponding problem for hypergraphs, for which less is known about the large deviation rate. We present new phenomena in upper tail large deviations for sparse random hypergraphs that are not seen in random graphs. We conjecture a formula for the large deviation rate, i.e., the first order asymptotics of the log-probability that the number of copies of fixed subgraph $H$ in a sparse Erdős--Rényi random $k$-uniform hypergraph exceeds its expectation by a constant factor. This conjecture turns out to be significantly more intricate compared to the case for graphs. We verify our conjecture when the fixed subgraph $H$ being counted is a clique, as well as when $H$ is the 3-uniform 6-vertex 4-edge hypergraph consisting of alternating faces of an octahedron, where new techniques are required.

Consider the random hypercube $H_2^n(p_n)$ obtained from the hypercube $H_2^n$ by deleting any given edge with probabilty $1-p_n$, independently of all the other edges. A diameter path in $H_2^n$ is a longest geodesic path in $H_2^n$. Consider the following two ways of tampering with the random graph $H_2^n(p_n)$: (i) choose a diameter path at random and adjoin all of its edges to $H_2^n(p_n)$; (ii) choose a diameter path at random from among those that start at $0=(0,..., 0)$, and adjoin all of its edges to $H_2^n(p_n)$. We study the question of whether these tamperings are detectable asymptotically as $n\to\infty$.

We consider an ensemble of nxn real symmetric random matrices A whose entries are determined by independent identically distributed random variables that have symmetric probability distribution. Assuming that the moment 12+2delta of these random variables exists, we prove that the probability distribution of the spectral norm of A rescaled to n^{-2/3} is bounded by a universal expression. The proof is based on the completed and modified version of the approach proposed and developed by Ya. Sinai and A. Soshnikov to study high moments of Wigner random matrices.

The study of longest increasing subsequences (LIS) in permutations led to that of Young diagrams via Robinson-Schensted's (RS) correspondence. In a celebrated paper, Vershik and Kerov obtained a limit theorem for such diagrams and found that the LIS of a uniform permutation of size n behaves as $2\sqrt{n}$. Independently and much later, Hoppen et al. introduced the theory of permutons as a scaling limit of permutations. In this paper, we extend in some sense the RS correspondence of permutations to the space of permutons. When the "RS-tableaux" of a permuton are non-trivial, we show that the RS-tableaux of random permutations sampled from this permuton exhibit a linear behavior, in the sense that their first rows and columns have lengths of linear order. In particular, the LIS of such permutations behaves as a multiple of n. We also prove some large deviation results for these convergences. Finally, by studying asymptotic properties of Fomin's algorithm for permutations, we show that the RS-tableaux of a permuton satisfy a partial differential equation.

We study the two-point functions of a general class of random-length random walks on finite boxes in $\ZZ^d$ with $d\ge3$, and provide precise asymptotics for their behaviour. We show that the finite-box two-point function is asymptotic to the infinite-lattice two-point function when the typical walk length is $o(L^2)$, but develops a plateau when the typical walk length is $Ω(L^2)$. We also numerically study walk length moments and limiting distributions of the self-avoiding walk and Ising model on five-dimensional tori, and find that they agree asymptotically with the known results for self-avoiding walk on the complete graph, both at the critical point and also for a broad class of scaling windows/pseudocritical points. Furthermore, we show that the two-point function of the finite-box random-length random walk, with walk length chosen via the complete graph self-avoiding walk, agrees numerically with the two-point functions of the self-avoiding walk and Ising model on five-dimensional tori. We conjecture that these observations in five dimensions should also hold in all higher dimensions.

We provide precise asymptotic estimates for the number of several classes of labelled cubic planar graphs, and we analyze properties of such random graphs under the uniform distribution. This model was first analyzed by Bodirsky et al. (Random Structures Algorithms 2007). We revisit their work and obtain new results on the enumeration of cubic planar graphs and on random cubic planar graphs. In particular, we determine the exact probability of a random cubic planar graph being connected, and we show that the distribution of the number of triangles in random cubic planar graphs is asymptotically normal with linear expectation and variance. To the best of our knowledge, this is the first time one is able to determine the asymptotic distribution for the number of copies of a fixed graph containing a cycle in classes of random planar graphs arising from planar maps.