LDP for Finite Dimensional Spaces.- Applications-The Case.- General Principles.- Sample Path Large Deviations.- The Abstract Empirical Measures.- Applications of Measures LDP.

Abstract We consider the extremes of logarithm characteristic polynomial matrices from C $\beta $ E ensemble. prove convergence in distribution centered maxima (of real and imaginary parts) toward sum a Gumbel variable another independent variable, which we characterize as total mass ‘derivative martingale’. also provide description landscape near extrema points.

We derive concentration inequalities for functions of the empirical measure eigenvalues large, random, self adjoint matrices, with not necessarily Gaussian entries. The results presented apply in particular to non-Gaussian Wigner and Wishart matrices. also provide bounds non commutative functionals random

Let T (x, ε) denote the first hitting time of disc radius ε centered at x for Brownian motion on two dimensional torus 2 .We prove that sup x∈T ε)/| log ε| → 2/π as 0. The same applies to any smooth, compact connected, two-dimensional, Riemannian manifold with unit area and no boundary.As a consequence, we conjecture, due Aldous (1989), number steps it takes simple random walk cover all points lattice Z n is asymptotic 4n (log n) /π.Determining these asymptotics an essential step toward...

We study the empirical measure LA n of eigenvalues nonnormal square matrices form An = UnTnVn with Un, Vn independent Haar distributed on unitary group and Tn real diagonal.We show that when converges, satisfies some technical conditions, converges towards a rotationally invariant µ complex plane whose support is single ring.In particular, we provide complete proof Feinberg-Zee ring theorem [6].We also consider case where are independently orthogonal group.

Abstract We consider the discrete two‐dimensional Gaussian free field on a box of side length $N$, with Dirichlet boundary data, and prove convergence law centered maximum field.© 2015 Wiley Periodicals, Inc.

Abstract We consider the maximum of discrete two‐dimensional Gaussian free field (GFF) in a box and prove that its maximum, centered at mean, is tight, settling longstanding conjecture. The proof combines recent observation by Bolthausen, Deuschel, Zeitouni with elements from Bramson's results on branching Brownian motion comparison theorems for fields. An essential part argument precise evaluation, up to an error order 1, expected value GFF box. Related fields, such as torus, are also...

Let |$U_N$| denote a Haar Unitary matrix of dimension |$N,$| and consider the field |$\mathbf{U}_{{N}}(z) = \log |\det(1-zU_N)|$| for |$z\in \mathbb{C}$|⁠. Then, |$\frac{\max_{|z|=1} \mathbf{U}_{{N}}(z) - N + \frac{3}{4} \log\log N}{ N} \to 0 $| in probability. This provides verification up to second order conjecture Fyodorov, Hiary, Keating, improving on recent first Arguin, Belius Bourgade.

Nakamoto invented the longest chain protocol, and claimed its security by analyzing private double-spend attack, a race between adversary honest nodes to grow longer chain. But is it worst attack? We answer question in affirmative for three classes of protocols, designed different consensus models: 1) Nakamoto's original Proof-of-Work protocol; 2) Ouroboros SnowWhite Proof-of-Stake protocols; 3) Chia Proof-of-Space protocol. As consequence, exact characterization maximum tolerable power...

We study the signal-to-interference (SIR) performance of linear multiuser receivers in random environments, where signals from users arrive "random directions." Such a environment may arise DS-CDMA system with signature sequences, or antenna diversity randomness is due to channel fading. Assuming that such directions can be tracked by receiver, resulting SIR function and therefore also random. asymptotic distribution this regime both number K degrees freedom N are large, but keeping their...

The generalized likelihood ratio test (GLRT), which is commonly used in composite hypothesis testing problems, investigated. Conditions for asymptotic optimality of the GLRT Neyman-Pearson sense are studied and discussed. First, a general necessary sufficient condition established, then based on this, condition, easier to verify, derived. A counterexample where not optimal, provided as well. conjecture stated concerning class finite-state sources.< <ETX...

Consider two independent sequences $X_1,\ldots, X_n$ and $Y_1,\ldots, Y_n$. Suppose that are i.i.d. $\mu_X$ Y_n$ $\mu_Y$, where $\mu_Y$ distributions on finite alphabets $\sigma_X$ $\sigma_Y$, respectively. A score $F: \sigma_X \times \sigma_Y\rightarrow \mathbb{R}$ is assigned to each pair $(X_i, Y_j)$ the maximal nonaligned segment $M_n = \max_{0\leq i, j\leq n - \Delta, \Delta \geq 0} \{\sum^\Delta_{k=1} F(X_{i+k}, Y_{j+k})\}$. The limit distribution of $M_n$ derived here when not too far...

Consider a graph with set of vertices and oriented edges connecting pairs vertices. Each vertex is associated random variable these are assumed to be independent. In this setting, suppose we wish solve the following hypothesis testing problem: under null, variables have common distribution N(0,1) while alternative, there an unknown path along which $N(μ,1)$, $μ&gt; 0$, away from it. For values mean shift $μ$ can one reliably detect for impossible? Consider, example, usual regular lattice...

Random walks in random environments and their diffusion analogues have been a source of surprising phenomena challenging problems, especially the non-reversible situation, since they began to be studied 1970s. We review model, available results techniques, point out several gaps understanding these processes.

We show that the centered maximum of a sequence logarithmically correlated Gaussian fields in any dimension converges distribution, under assumption covariances converge suitable sense. identify limit as randomly shifted Gumbel and characterize random shift distribution variables, reminiscent derivative martingale theory branching walk chaos. also discuss applications main convergence theorem examples for fields; some additional structural assumptions type we make are needed maximum.

and conjectured that the limit exists equals 1/Tr a.s. The importance of determining value this is clarified in (1.3) below, where appears power laws governing local time walk. Erd6s-Taylor conjecture was quoted book by R~v~sz [19, w but to best our knowledge, bounds (1.1) were not improved prior present paper. As it turns out, an important step towards solution formulation Perkins Taylor [17] analogous problem on maximal occupation measure planar Brownian motion (run for unit time) can