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Let W N (β) = E 0 e ∑ n = 1 N β ω (n , S n) - N β 2 / 2 be the partition function of a two-dimensional directed polymer in a random environment, where ω (i , x) , i ∈ N , x ∈ Z 2 are i.i.d. standard normal and { S n } is the path of a random walk. With β = β N = β ^ π / log N and β ^ ∈ (0 , 1) (the subcritical window), log W N (β N) is known to converge in distribution to a Gaussian law of mean - λ 2 / 2 and variance λ 2 , with λ 2 = log (1 / (1 - β ^ 2)) (Caravenna et al. in Ann Appl Probab 27(5):3050–3112, 2017). We study in this paper the moments E [ W N (β N) q ] in the subcritical window, for q = O (log N) . The analysis is based on ruling out triple intersections. [ABSTRACT FROM AUTHOR]

We prove a central limit theorem for the real part of the logarithm of the characteristic polynomial of random Jacobi matrices. Our results cover the G β E models for β > 0 . [ABSTRACT FROM AUTHOR]

We prove that Ising models on the hypercube with general quadratic interactions satisfy a Poincaré inequality with respect to the natural Dirichlet form corresponding to Glauber dynamics, as soon as the operator norm of the interaction matrix is smaller than 1. The inequality implies a control on the mixing time of the Glauber dynamics. Our techniques rely on a localization procedure which establishes a structural result, stating that Ising measures may be decomposed into a mixture of measures with quadratic potentials of rank one, and provides a framework for proving concentration bounds for high temperature Ising models. [ABSTRACT FROM AUTHOR]

We consider the stochastic heat equation ∂ s u = 1 2 Δ u + (β V (s , y) - λ) u , with a smooth space-time-stationary Gaussian random field V(s, y), in dimensions d ≥ 3 , with an initial condition u (0 , x) = u 0 (ε x) and a suitably chosen λ ∈ R . It is known that, for β small enough, the diffusively rescaled solution u ε (t , x) = u (ε - 2 t , ε - 1 x) converges weakly to a scalar multiple of the solution u ¯ (t , x) of the heat equation with an effective diffusivity a, and that fluctuations converge, also in a weak sense, to the solution of the Edwards-Wilkinson equation with an effective noise strength ν and the same effective diffusivity. In this paper, we derive a pointwise approximation w ε (t , x) = u ¯ (t , x) Ψ ε (t , x) + ε u 1 ε (t , x) , where Ψ ε (t , x) = Ψ (t / ε 2 , x / ε) , Ψ is a solution of the SHE with constant initial conditions, and u 1 ε is an explicit corrector. We show that Ψ (t , x) converges to a stationary process Ψ ~ (t , x) as t → ∞ , that E | u ε (t , x) - w ε (t , x) | 2 converges pointwise to 0 as ε → 0 , and that ε - d / 2 + 1 (u ε - w ε) converges weakly to 0 for fixed t. As a consequence, we derive new representations of the diffusivity a and effective noise strength ν . Our approach uses a Markov chain in the space of trajectories introduced in [17], as well as tools from homogenization theory. The corrector u 1 ε (t , x) is constructed using a seemingly new approximation scheme on a mesoscopic time scale. [ABSTRACT FROM AUTHOR]

We prove that the minimum of the modulus of a random trigonometric polynomial with Gaussian coefficients, properly normalized, has limiting exponential distribution. [ABSTRACT FROM AUTHOR]

We study the directed polymer model for general graphs (beyond Z d ) and random walks. We provide sufficient conditions for the existence or non-existence of a weak disorder phase, of an L 2 region, and of very strong disorder, in terms of properties of the graph and of the random walk. We study in some detail (biased) random walk on various trees including the Galton–Watson trees, and provide a range of other examples that illustrate counter-examples to intuitive extensions of the Z d /SRW result. [ABSTRACT FROM AUTHOR]

Let { S k : k ≥ 0 } be a symmetric and aperiodic random walk on Z d , d ≥ 3 , and { ξ (z) , z ∈ Z d } a collection of independent and identically distributed random variables. Consider a random walk in random scenery defined by T n = ∑ k = 0 n ξ (S k) = ∑ z ∈ Z d l n (z) ξ (z) , where l n (z) = ∑ k = 0 n I { S k = z } is the local time of the random walk at the site z. Using (∑ z ∈ Z d l n (z) | ξ (z) | p ) 1 / p , p ≥ 2 , as the normalizing constants, we establish self-normalized moderate deviations for random walk in random scenery under a much weaker condition than a finite moment-generating function of the scenery variables. [ABSTRACT FROM AUTHOR]

Consider an N × N Toeplitz matrix T N with symbol a (λ) : = ∑ ℓ = - d 2 d 1 a ℓ λ ℓ , perturbed by an additive noise matrix N - γ E N , where the entries of E N are centered i.i.d. random variables of unit variance and γ > 1 / 2 . It is known that the empirical measure of eigenvalues of the perturbed matrix converges weakly, as N → ∞ , to the law of a (U) , where U is distributed uniformly on S 1 . In this paper, we consider the outliers, i.e. eigenvalues that are at a positive (N-independent) distance from a (S 1) . We prove that there are no outliers outside spec T (a) , the spectrum of the limiting Toeplitz operator, with probability approaching one, as N → ∞ . In contrast, in spec T (a) \ a (S 1) the process of outliers converges to the point process described by the zero set of certain random analytic functions. The limiting random analytic functions can be expressed as linear combinations of the determinants of finite sub-matrices of an infinite dimensional matrix, whose entries are i.i.d. having the same law as that of E N . The coefficients in the linear combination depend on the roots of the polynomial P z , a (λ) : = (a (λ) - z) λ d 2 and semi-standard Young Tableaux with shapes determined by the number of roots of P z , a (λ) = 0 that are greater than one in moduli. [ABSTRACT FROM AUTHOR]

Lemma 9.1 in the paper is incorrect as stated. [ABSTRACT FROM AUTHOR]

Let C ϵ , S 2 ∗ denote the cover time of the two dimensional sphere by a Wiener sausage of radius ϵ . We prove that C ϵ , S 2 ∗ - 2 A S 2 π log ϵ - 1 - 1 4 log log ϵ - 1 is tight, where A S 2 = 4 π denotes the Riemannian area of S 2 . [ABSTRACT FROM AUTHOR]

We prove, using probabilistic techniques and analysis on the Wiener space, that the large scale fluctuations of the KPZ equation in d ≥ 3 with a small coupling constant, driven by a white in time and colored in space noise, are given by the Edwards-Wilkinson model. This gives an alternative proof, that avoids perturbation expansions, to the results of Magnen and Unterberger (J Stat Phys 171:543–598, 2018). [ABSTRACT FROM AUTHOR]

We show the existence of the scaling exponent χ = χ (γ) , with 0 < χ ≤ 4 γ 2 1 + γ 2 / 4 - 1 + γ 4 / 16 , of the graph distance associated with subcritical two-dimensional Liouville quantum gravity of paramater γ < 2 on V = [ 0 , 1 ] 2 . We also show that the Liouville heat kernel satisfies, for each fixed u , v ∈ V o , the short time estimate lim t → 0 log | log p t γ (u , v) | | log t | = χ 2 - χ , a. s.. [ABSTRACT FROM AUTHOR]

We consider the heat equation with a multiplicative Gaussian potential in dimensions d ≥ 3. We show that the renormalized solution converges to the solution of a deterministic diffusion equation with an effective diffusivity. We also prove that the renormalized large scale random fluctuations are described by the Edwards-Wilkinson model, that is, the stochastic heat equation (SHE) with additive white noise, with an effective variance. [ABSTRACT FROM AUTHOR]

We study the partition function of the Curie-Weiss model with complex temperature, and partially describe its phase transitions. As a consequence, we obtain information on the locations of zeros of the partition function (the Fisher zeros). [ABSTRACT FROM AUTHOR]

Recently, sharp results concerning the critical points of the Hamiltonian of the p-spin spherical spin glass model have been obtained by means of moments computations. In particular, these moments computations allow for the evaluation of the leading term of the ground-state, i.e., of the global minimum. In this paper, we study the extremal point process of critical points-that is, the point process associated to all critical values in the vicinity of the ground-state. We show that the latter converges in distribution to a Poisson point process of exponential intensity. In particular, we identify the correct centering of the ground-state and prove the convergence in distribution of the centered minimum to a (minus) Gumbel variable. These results are identical to what one obtains for a sequence of i.i.d variables, correctly normalized; namely, we show that the model is in the universality class of REM. [ABSTRACT FROM AUTHOR]

We derive a large deviations principle for the two-dimensional two-component plasma in a box. As a consequence, we obtain a variational representation for the free energy, and also show that the macroscopic empirical measure of either positive or negative charges converges to the uniform measure. An appendix, written by Wei Wu, discusses applications to the subcritical complex Gaussian multiplicative chaos. [ABSTRACT FROM AUTHOR]

We consider random polynomials whose coefficients are independent and uniform on {-1, 1}. We prove that the probability that such a polynomial of degree n has a double root is o( n ) when n+1 is not divisible by 4 and asymptotic to $$1/\sqrt 3 $$ otherwise. This result is a corollary of a more general theorem that we prove concerning random polynomials with independent, identically distributed coefficients having a distribution which is supported on {-1, 0, 1} and whose largest atom is strictly less than $$\frac{{8\sqrt 3 }}{{\pi {n^2}}}$$ . In this general case, we prove that the probability of having a double root equals the probability that either -1, 0 or 1 are double roots up to an o( n ) factor and we find the asymptotics of the latter probability. [ABSTRACT FROM AUTHOR]

The limiting extremal processes of the branching Brownian motion (BBM), the two-speed BBM, and the branching random walk are known to be randomly shifted decorated Poisson point processes (SDPPP). In the proofs of those results, the Laplace functional of the limiting extremal process is shown to satisfy $${L\left[\theta_{y}f\right]=g\left(y-\tau_{f}\right)}$$ for any nonzero, nonnegative, compactly supported, continuous function f, where $${\theta_{y}}$$ is the shift operator, $${\tau_{f}}$$ is a real number that depends on f, and g is a real function that is independent of f. We show that, under some assumptions, this property characterizes the structure of SDPPP. Moreover, when it holds, we show that g has to be a convolution of the Gumbel distribution with some measure. The above property of the Laplace functional is closely related to a 'freezing phenomenon' that is expected to occur in a wide class of log-correlated fields, and which has played an important role in the analysis of various models. Our results shed light on this intriguing phenomenon and provide a natural tool for proving an SDPPP structure in these and other models. [ABSTRACT FROM AUTHOR]

We consider the quadratic optimization problem with $$W$$ a (random) matrix and $$\mathbf{h}$$ a random external field. We study the probabilities of large deviation of $$F_n^{W,\mathbf{h}}$$ for $$\mathbf{h}$$ a centered Gaussian vector with i.i.d. entries, both conditioned on $$W$$ (a general Wigner matrix), and unconditioned when $$W$$ is a GOE matrix. Our results validate (in a certain region) and correct (in another region), the prediction obtained by the mathematically non-rigorous replica method in Fyodorov and Doussal (J Stat Phys 154:466-490, ). [ABSTRACT FROM AUTHOR]

We study the limiting distribution of the eigenvalues of the Ginibre ensemble conditioned on the event that a certain proportion lie in a given region of the complex plane. Using an equivalent formulation as an obstacle problem, we describe the optimal distribution and some of its monotonicity properties. [ABSTRACT FROM AUTHOR]

We study the eigenvalues of non-normal square matrices of the form A = U T V with U, V independent Haar distributed on the unitary group and T real diagonal. We show that when the empirical measure of the eigenvalues of T converges, and T satisfies some technical conditions, all these eigenvalues lie in a single ring. [ABSTRACT FROM AUTHOR]

We consider the maximal displacement of one dimensional branching Brownian motion with (macroscopically) time varying profiles. For monotone decreasing variances, we show that the correction from linear displacement is not logarithmic but rather proportional to T. We conjecture that this is the worse case correction possible. [ABSTRACT FROM AUTHOR]

We consider random walks in a balanced random environment in $${\mathbb{Z}^d}$$, d ≥ 2. We first prove an invariance principle (for d ≥ 2) and the transience of the random walks when d ≥ 3 (recurrence when d = 2) in an ergodic environment which is not uniformly elliptic but satisfies certain moment condition. Then, using percolation arguments, we show that under mere ellipticity, the above results hold for random walks in i.i.d. balanced environments. [ABSTRACT FROM AUTHOR]

We consider the quenched and the averaged (or annealed) large deviation rate functions I and I for space-time and (the usual) space-only RWRE on $${\mathbb{Z}^d}$$ . By Jensen's inequality, I ≤ I. In the space-time case, when d ≥ 3 + 1, I and I are known to be equal on an open set containing the typical velocity ξ. When d = 1 + 1, we prove that I and I are equal only at ξ. Similarly, when d = 2 + 1, we show that I < I on a punctured neighborhood of ξ. In the space-only case, we provide a class of non-nestling walks on $${\mathbb{Z}^d}$$ with d = 2 or 3, and prove that I and I are not identically equal on any open set containing ξ whenever the walk is in that class. This is very different from the known results for non-nestling walks on $${\mathbb{Z}^d}$$ with d ≥ 4. [ABSTRACT FROM AUTHOR]

Let $${\mathcal{T}}$$ be a rooted Galton–Watson tree with offspring distribution { p k } that has p 0 = 0, mean m = ∑ kp k > 1 and exponential tails. Consider the λ-biased random walk { X n } n ≥ 0 on $${\mathcal{T}}$$ ; this is the nearest neighbor random walk which, when at a vertex v with d v offspring, moves closer to the root with probability λ/(λ + d v ), and moves to each of the offspring with probability 1/(λ + d v ). It is known that this walk has an a.s. constant speed $${\tt v} = \lim_n |X_n|/n$$ (where | X n | is the distance of X n from the root), with $${\tt v} > 0$$ for 0 < λ < m and $${\tt v} = 0$$ for λ ≥ m. For all λ ≤ m, we prove a quenched CLT for $$|X_n| - n{\tt v}$$ . (For λ > m the walk is positive recurrent, and there is no CLT.) The most interesting case by far is λ = m, where the CLT has the following form: for almost every $${\mathcal{T}}$$ , the ratio $$|X_{[nt]}|/\sqrt{n}$$ converges in law as n → ∞ to a deterministic multiple of the absolute value of a Brownian motion. Our approach to this case is based on an explicit description of an invariant measure for the walk from the point of view of the particle (previously, such a measure was explicitly known only for λ = 1) and the construction of appropriate harmonic coordinates. [ABSTRACT FROM AUTHOR]

A quenched central limit theorem is derived for the super-Brownian motion with super-Brownian immigration, in dimension d≥4. At the critical dimension d=4, the quenched and annealed fluctuations are of the same order but are not equal. [ABSTRACT FROM AUTHOR]

We present a multiscale analysis for the exit measures from large balls in $$\mathbb{Z}^d, d\geq 3$$ , of random walks in certain i.i.d. random environments which are small perturbations of the fixed environment corresponding to simple random walk. Our main assumption is an isotropy assumption on the law of the environment, introduced by Bricmont and Kupiainen. Under this assumption, we prove that the exit measure of the random walk in a random environment from a large ball, approaches the exit measure of a simple random walk from the same ball, in the sense that the variational distance between smoothed versions of these measures converges to zero. We also prove the transience of the random walk in random environment. The analysis is based on propagating estimates on the variational distance between the exit measure of the random walk in random environment and that of simple random walk, in addition to estimates on the variational distance between smoothed versions of these quantities. [ABSTRACT FROM AUTHOR]

We investigate in this work the asymptotic behavior of isotropic diffusions in random environment that are small perturbations of Brownian motion. When the space dimension is three or more, we prove an invariance principle as well as transience. Our methods also apply to questions of homogenization in random media. [ABSTRACT FROM AUTHOR]

A law of large numbers and a central limit theorem are derived for linear statistics of random symmetric matrices whose on-or-above diagonal entries are independent, but neither necessarily identically distributed, nor necessarily all of the same variance. The derivation is based on systematic combinatorial enumeration, study of generating functions, and concentration inequalities of the Poincaré type. Special cases treated, with an explicit evaluation of limiting variances, are generalized Wigner and Wishart matrices. [ABSTRACT FROM AUTHOR]

We consider a class of ballistic, multidimensional random walks in random environments where the environment satisfies appropriate mixing conditions. Continuing our previous work [2] for the law of large numbers, we prove here that the fluctuations are Gaussian when the environment is Gibbsian satisfying the “strong mixing condition” of Dobrushin and Shlosman and the mixing rate is large enough to balance moments of some random times depending on the path. Under appropriate assumptions the annealed Central Limit Theorem (CLT) applies in both nonnestling and nestling cases, and trivially in the case of finite-dependent environments with “strong enough bias”. Our proof makes use of the asymptotic regeneration scheme introduced in [2]. When the environment is only weakly mixing, we can only prove that if the fluctuations are diffusive then they are necessarily Gaussian. [ABSTRACT FROM AUTHOR]

Studies large deviations for random walks in random environment. Key distinction between quenched asymptotics, conditional on the environment and annealed asymptotics, obtained from averaging over environments; Simple random walk on Galton-Watson family tree, arising from a supercritical branching process; Uncertainty in the location of the hitting point.

Suppose that the integers are assigned random variables ω[sub i] (taking values in the unit interval), which serve as an environment. This environment defines a random walk X[sub n] (called a RWRE) which, when at i, moves one step to the right with probability ω[sub i] , and one step to the left with probability 1 - ω[sub i] . When the ω[sub i] sequence is i.i.d., Greven and den Hollander (1994) proved a large deviation principle for X[sub n] /n, conditional upon the environment, with deterministic rate function. We consider in this paper large deviations, both conditioned on the environment (quenched) and averaged on the environment (annealed), for the RWRE, in the ergodic environment case. The annealed rate function is the solution of a variational problem involving the quenched rate function and specific relative entropy. We also give a detailed qualitative description of the resulting rate functions. Our techniques differ from those of Greven and den Hollander, and allow us to present also a trajectorial (quenched) large deviation principle. [ABSTRACT FROM AUTHOR]

Abstract. Suppose that the integers are assigned i.i.d, random variables {omega[sub x]} (taking values in the interval [1/2, 1)), which serve as an environment. This environment defines a random walk {X[sub k]} (called a RWRE) which, when at x, moves one step to the right with probability omega[sub x], and one step to the left with probability 1 - omega[sub x]. Solomon (1975) determined the almost-sure asymptotic speed (= rate of escape) of a RWRE, in a more general set-up. Dembo, Peres and Zeitouni (1996), following earlier work by Greven and den Hollander (1994) on the quenched case, have computed rough tail asymptotics for the empirical mean of the annealed RWRE. They conjectured the form of the rate function in a full LDP. We prove in this paper their conjecture. The proof is based on a "coarse graining scheme" together with comparison techniques. [ABSTRACT FROM AUTHOR]

An Onsager Machlup functional limit is derived for a class of SPDE's whose principal part is not trace class. Both nondegenerate and degenerate limits are obtained, and are illustrated by examples. The proof uses FKG type inequalities. [ABSTRACT FROM AUTHOR]

Suppose that the integers are assigned i.i.d. random variables {ωx} (taking values in the unit interval), which serve as an environment. This environment defines a random walk { X n } (called a RWRE) which, when at x, moves one step to the right with probability ωx, and one step to the left with probability 1- ω x . Solomon (1975) determined the almost-sure asymptotic speed v α (=rate of escape) of a RWRE. Greven and den Hollander (1994) have proved a large deviation principle for X n / n, conditional upon the environment, with deterministic rate function. For certain environment distributions where the drifts 2 ω x -1 can take both positive and negative values, their rate function vanisheson an interval (0, v α). We find the rate of decay on this interval and prove it is a stretched exponential of appropriate exponent, that is the absolute value of the log of the probability that the empirical mean X n / n is smaller than v, v∈ (0, v α), behaves roughly like a fractional power of n. The annealed estimates of Dembo, Peres and Zeitouni (1996) play a crucial role in the proof. We also deal with the case of positive and zero drifts, and prove there a quenched decay of the form . [ABSTRACT FROM AUTHOR]

Suppose that the integers are assigned i.i.d. random variables {ω} (taking values in the unit interval), which serve as an environment. This environment defines a random walk { X} (called a RWRE) which, when at x, moves one step to the right with probability ω, and one step to the left with probability 1-ω. Solomon (1975) determined the almost-sure asymptotic speed (=rate of escape) of a RWRE. For certain environment distributions where the drifts 2ω-1 can take both positive and negative values, we show that the chance of the RWRE deviating below this speed has a polynomial rate of decay, and determine the exponent in this power law; for environments which allow only positive and zero drifts, we show that these large-deviation probabilities decay like exp(− Cn). This differs sharply from the rates derived by Greven and den-Hollander (1994) for large deviation probabilities conditioned on the environment. As a by product we also provide precise tail and moment estimates for the total population size in a Branching Process with Random Environment. [ABSTRACT FROM AUTHOR]