[Show abstract][Hide abstract] ABSTRACT: Suppose the auto-correlations of real-valued, centered Gaussian process
$Z(\cdot)$ are non-negative and decay as $\rho(|s-t|)$ for some $\rho(\cdot)$
regularly varying at infinity of order $-\alpha \in [-1,0)$. With
$I_\rho(t)=\int_0^t \rho(s)ds$ its primitive, we determine the precise
persistence probabilities decay rate $$ -\log\mathbb{P}(\sup_{t \in
[0,T]}\{Z(t)\}<0)=\Theta\Big(\frac{T\log I_\rho(T)}{I_\rho(T)}\Big) \,, $$
thereby closing the gap between the lower and upper bounds of {NR}, which stood
as such for over fifty years. We demonstrate its usefulness by sharpening
recent results of {Sak} about the dependence on $d$ of such persistence decay
for the Langevin dynamics of certain $\nabla \phi$-interface models on
$\mathbb{Z}^d$.
[Show abstract][Hide abstract] ABSTRACT: We extend the use of random evolving sets to time-varying conductance models
and utilize it to provide tight heat kernel upper bounds. It yields the
transience of any uniformly lazy random walk, on Z^d, d>=3, equipped with
uniformly bounded above and below, independently time-varying edge
conductances, having non-decreasing in time vertex conductances (i.e. reversing
measure), thereby affirming part of [ABGK, Conj. 7.1].
[Show abstract][Hide abstract] ABSTRACT: We study the fluctuation of the Atlas model, where a unit drift is assigned
to the lowest ranked particle among a semi-infinite ($ \mathbb{Z}_+ $-indexed)
system of otherwise independent Brownian particles, initiated according to a
Poisson point process on $ \mathbb{R}_+ $. In this context, we show that the
joint law of ranked particles, after being centered and scaled by $t^{-1/4}$,
converges as $t \to \infty$ to the Gaussian field corresponding to the solution
of the additive stochastic heat equation on $\mathbb{R}_+$ with Neumann
boundary condition at zero. This allows us to express the asymptotic
fluctuation of the lowest ranked particle in terms of a $ \frac{1}{4}
$-fractional Brownian motion. In particular, we prove a conjecture of Pal and
Pitman (2008) about the asymptotic Gaussian fluctuation of the ranked
particles.
[Show abstract][Hide abstract] ABSTRACT: For Erd\H{o}s-R\'enyi random graphs with average degree $\gamma$, and
uniformly random $\gamma$-regular graph on $n$ vertices, we prove that with
high probability the size of both the Max-Cut and maximum bisection are
$n\Big(\frac{\gamma}{4} + {{\sf P}}_* \sqrt{\frac{\gamma}{4}} +
o(\sqrt{\gamma})\Big) + o(n)$ while the size of the minimum bisection is
$n\Big(\frac{\gamma}{4}-{{\sf P}}_*\sqrt{\frac{\gamma}{4}} +
o(\sqrt{\gamma})\Big) + o(n)$. Our derivation relates the free energy of the
anti-ferromagnetic Ising model on such graphs to that of the
Sherrington-Kirkpatrick model, with ${{\sf P}}_* \approx 0.7632$ standing for
the ground state energy of the latter, expressed analytically via Parisi's
formula.
[Show abstract][Hide abstract] ABSTRACT: We consider the quadratic optimization problem $$F_n^{W,h}:= \sup_{x \in
S^{n-1}} ( x^T W x/2 + h^T x )\,, $$ with $W$ a (random) matrix and $h$ a
random external field. We study the probabilities of large deviation of
$F_n^{W,h}$ for $h$ 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 Y. V. Fyodorov, P. Le Doussal, J. Stat. phys. 154 (2014).
[Show abstract][Hide abstract] ABSTRACT: We consider recurrence versus transience for models of random walks on
domains of $\mathbb{Z}^d$, in which monotone interaction enforces domain growth
as a result of visits by the walk (or probes it sent), to the neighborhood of
domain boundary.
Electronic communications in probability 06/2014; 19. DOI:10.1214/ECP.v19-3607 · 0.62 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We present a general technique for computing large deviations of nonlinear
functions of independent Bernoulli random variables. The method is applied to
compute the large deviation rate functions for subgraph counts in sparse random
graphs. Previous technology, based on Szemer\'edi's regularity lemma, works
only for dense graphs. Applications are also made to exponential random graphs
and three-term arithmetic progressions in random sets of integers.
[Show abstract][Hide abstract] ABSTRACT: For normally reflected Brownian motion and for simple random walk on growing
in time d-dimensional domains, d>=3, we establish a sharp criterion for
recurrence versus transience in terms of the growth rate.
ELECTRONIC JOURNAL OF PROBABILITY 12/2013; 19. DOI:10.1214/EJP.v19-3272 · 0.77 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Suppose that ${\mathcal G}$ is a finite, connected graph and $X$ is a lazy
random walk on ${\mathcal G}$. The lamplighter chain $X^\diamond$ associated
with $X$ is the lazy random walk on the wreath product ${\mathcal G}^\diamond =
{\mathbb Z}_2 \wr {\mathcal G}$, the graph whose vertices consist of pairs
$(\underline{f},x)$ where $\underline{f}$ is a $\{0,1\}$-labeling of the
vertices of ${\mathcal G}$ and $x$ is a vertex in ${\mathcal G}$. In each step,
$X^\diamond$ moves from a configuration $(\underline{f},x)$ by updating $x$ to
$y$ using the transition rule of $X$, and if $x \ne y$, replacing $f_x$ and
$f_y$ by two independent uniform random bits. The mixing time of the
lamplighter chain on the discrete torus ${\mathbb Z}_n^d$ is known to have a
cutoff at a time asymptotic to the cover time of ${\mathbb Z}_n^d$ if $d=2$,
and to half the cover time if $d \geq 3$. We show that the mixing time of the
lamplighter chain on ${\mathcal G}_n(a)={\mathbb Z}_n^2 \times {\mathbb Z}_{a
\log n}$ has a cutoff at $\psi(a)$ times the cover time of ${\mathcal G}_n(a)$
as $n \to \infty$, where $\psi$ is a weakly decreasing map from $(0,\infty)$
onto $[1/2,1)$. In particular, as $a > 0$ varies, the threshold continuously
interpolates between the known thresholds for ${\mathbb Z}_n^2$ and ${\mathbb
Z}_n^3$. Perhaps surprisingly, we find a phase transition (non-smoothness of
$\psi$) at the point $a_*=\pi r_3 (1+\sqrt{2})$, where high dimensional
behavior ($\psi(a)=1/2$ for all $a>a_*$) commences. Here $r_3$ is the effective
resistance from $0$ to $\infty$ in ${\mathbb Z}^3$.
[Show abstract][Hide abstract] ABSTRACT: In this short note, we present a theorem concerning certain "additive
structure" for the level sets of non-degenerate Gaussian fields, which yields
the multiple valley phenomenon for extremal fields with exponentially many
valleys.
[Show abstract][Hide abstract] ABSTRACT: We analyze a class of non-simple exclusion processes and the corresponding
growth models by generalizing the Gaertner transformation. While the original
argument, which matches three identities with three parameters, applies only to
simple exclusion process, our approach is to identify the major non-linear
drift term of the microscopic dynamical equation and to convert the drift term
into a quasi-linear second order differential, yielding the stochastic heat
equation (SHE) in the continuum limit. Using the generalized transformation, we
prove convergence toward the KPZ equation, which is the first universality
result of this kind in the context of KPZ universality class. While this class
of exclusion processes are not explicitly solvable, we obtain the exact
one-point limit distribution for the step and step Bernoulli initial conditions
by using the previous results of Amir et al. (2011), Corwin & Quastel (2010),
and our convergence result.
[Show abstract][Hide abstract] ABSTRACT: We prove a Large Deviations Principle (LDP) for systems of diffusions
(particles) interacting through their ranks, when the number of particles tends
to infinity. We show that the limiting particle density is given by the unique
solution of the approriate McKean-Vlasov equation and that the corresponding
cumulative distribution function evolves according to the porous medium
equation with convection. The large deviations rate function is provided in
explicit form. This is the first instance of a LDP for interacting diffusions,
where the interaction occurs both through the drift and the diffusion
coefficients and where the rate function can be given explicitly. In the course
of the proof, we obtain new regularity results for a certain tilted version of
the porous medium equation.
[Show abstract][Hide abstract] ABSTRACT: We show that the empirical eigenvalue measure for sum of $d$ independent Haar
distributed $n$-dimensional unitary matrices, converge for $n \to \infty$ to
the Brown measure of the free sum of $d$ Haar unitary operators. The same
applies for independent Haar distributed $n$-dimensional orthogonal matrices.
As a byproduct of our approach, we relax the requirement of uniformly bounded
imaginary part of Stieltjes transform of $T_n$ that is made in [Guionnet,
Krishnapur, Zeitouni; Theorem 1].
Electronic communications in probability 08/2012; 18. DOI:10.1214/ECP.v18-2466 · 0.62 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Consider random polynomial $\sum_{i=0}^n a_i x^i$ of independent mean-zero
normal coefficients $a_i$, whose variance is a regularly varying function (in
$i$) of order $\alpha$. We derive general criteria for continuity of
persistence exponents for centered Gaussian processes, and use these to show
that such polynomial has no roots in $[0,1]$ with probability
$n^{-b_\alpha+o(1)}$, and no roots in $(1,\infty)$ with probability
$n^{-b_0+o(1)}$, hence for $n$ even, it has no real roots with probability
$n^{-2b_\alpha - 2b_0+o(1)}$. Here $b_\alpha =0$ when $\alpha \le -1$ and
otherwise $b_\alpha \in (0,\infty)$ is independent of the detailed regularly
varying variance function and corresponds to persistence probabilities for an
explicit stationary Gaussian process of smooth sample path. Further, making
precise the solution $\phi_d(x,t)$ to the $d$-dimensional heat equation
initiated by a Gaussian white noise $\phi_d(x,0)$, we confirm that the
probability of $\phi_d(x,t)\neq 0$ for all $t\in [1,T]$, is $T^{-b_{\alpha} +
o(1)}$, for $\alpha=d/2-1$.
The Annals of Probability 08/2012; 43(1). DOI:10.1214/13-AOP852 · 1.42 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We provide an explicit formula for the limiting free energy density
(log-partition function divided by the number of vertices) for ferromagnetic
Potts models on uniformly sparse graph sequences converging locally to the
d-regular tree for d even, covering all temperature regimes. This formula
coincides with the Bethe free energy functional evaluated at a suitable fixed
point of the belief propagation recursion on the d-regular tree, the so-called
replica symmetric solution. For uniformly random d-regular graphs we further
show that the replica symmetric Bethe formula is an upper bound for the
asymptotic free energy for any model with permissive interactions.
[Show abstract][Hide abstract] ABSTRACT: We consider the ferromagnetic Ising model on a sequence of graphs $G_n$
converging locally weakly to a rooted random tree. Generalizing [Montanari,
Mossel, Sly '11], under an appropriate "continuity" property, we show that the
Ising measures on these graphs converge locally weakly to a measure, which is
obtained by first picking a random tree, and then the symmetric mixture of
Ising measures with + and - boundary conditions on that tree. Under the extra
assumptions that $G_n$, of uniformly bounded degrees, are edge-expanders, and
ergodicity of the simple random walk on the limiting tree, we show that the
local weak limit of the Ising measures conditioned on positive magnetization,
is the Ising measure with + boundary condition on the limiting tree. We confirm
the "continuity" and ergodicity properties in case of limiting (multi-type)
Galton Watson trees, and the edge-expander property for the corresponding
configuration model graphs.
[Show abstract][Hide abstract] ABSTRACT: Let $S_n^{(2)}$ denote the iterated partial sums. That is,
$S_n^{(2)}=S_1+S_2+ ... +S_n$, where $S_i=X_1+X_2+ ... s+X_i$. Assuming $X_1,
X_2,....,X_n$ are integrable, zero-mean, i.i.d. random variables, we show that
the persistence probabilities $$p_n^{(2)}:=\PP(\max_{1\le i \le n}S_i^{(2)}< 0)
\le c\sqrt{\frac{\EE|S_{n+1}|}{(n+1)\EE|X_1|}},$$ with $c \le 6 \sqrt{30}$ (and
$c=2$ whenever $X_1$ is symmetric). The converse inequality holds whenever the
non-zero $\min(-X_1,0)$ is bounded or when it has only finite third moment and
in addition $X_1$ is squared integrable. Furthermore, $p_n^{(2)}\asymp
n^{-1/4}$ for any non-degenerate squared integrable, i.i.d., zero-mean $X_i$.
In contrast, we show that for any $0 < \gamma < 1/4$ there exist integrable,
zero-mean random variables for which the rate of decay of $p_n^{(2)}$ is
$n^{-\gamma}$.
[Show abstract][Hide abstract] ABSTRACT: We consider homogeneous factor models on uniformly sparse graph sequences
converging locally to a (unimodular) random tree T, and study the existence of
the free energy density phi, the limit of the log-partition function divided by
the number of vertices n as n tends to infinity. We provide a new interpolation
scheme and use it to prove existence of, and to explicitly compute, the
quantity phi subject to uniqueness of a relevant Gibbs measure for the factor
model on T. By way of example we compute phi for the independent set (or
hard-core) model at low fugacity, for the ferromagnetic Ising model at all
parameter values, and for the ferromagnetic Potts model with both weak enough
and strong enough interactions. Even beyond uniqueness regimes our
interpolation provides useful explicit bounds on phi.
In the regimes in which we establish existence of the limit, we show that it
coincides with the Bethe free energy functional evaluated at a suitable fixed
point of the belief propagation (Bethe) recursions on T. In the special case
that T has a Galton-Watson law, this formula coincides with the non-rigorous
"Bethe prediction" obtained by statistical physicists using the "replica" or
"cavity" methods. Thus our work is a rigorous generalization of these heuristic
calculations to the broader class of sparse graph sequences converging locally
to trees. We also provide a variational characterization for the Bethe
prediction in this general setting, which is of independent interest.
The Annals of Probability 10/2011; 41(6). DOI:10.1214/12-AOP828 · 1.42 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Let p_n denote the persistence probability that the first n iterated partial
sums of integrable, zero-mean, i.i.d. random variables X_k, are negative. We
show that p_n is bounded above up to universal constant by the square root of
the expected absolute value of the empirical average of {X_k}. A converse bound
holds whenever P(-X_1>t) is up to constant exp(-b t) for some b>0 or when
P(-X_1>t) decays super-exponentially in t. Consequently, for such random
variables we have that p_n decays as n^{-1/4} if X_1 has finite second moment.
In contrast, we show that for any 0 < c < 1/4 there exist integrable, zero-mean
random variables for which the rate of decay of p_n is n^{-c}.
[Show abstract][Hide abstract] ABSTRACT: Let T be a rooted supercritical multi-type Galton-Watson (MGW) tree with
types coming from a finite alphabet, conditioned to non-extinction. The
lambda-biased random walk (X_t, t>=0) on T is the nearest-neighbor random walk
which, when at a vertex v with d(v) offspring, moves closer to the root with
probability lambda/[lambda+d(v)], and to each of the offspring with probability
1/[lambda+d(v)]. This walk is recurrent for lambda>=rho and transient for
0<lambda<rho, with rho the Perron-Frobenius eigenvalue for the (assumed)
irreducible matrix of expected offspring numbers. Subject to finite moments of
order p>4 for the offspring distributions, we prove the following quenched CLT
for lambda-biased random walk at the critical value lambda=rho: for almost
every T, the process |X_{floor(nt)}|/sqrt{n} converges in law as n tends to
infinity to a reflected Brownian motion rescaled by an explicit constant. This
result was proved under some stronger assumptions by Peres-Zeitouni (2008) for
single-type Galton-Watson trees. Following their approach, our proof is based
on a new explicit description of a reversing measure for the walk from the
point of view of the particle (generalizing the measure constructed in the
single-type setting by Peres-Zeitouni), and the construction of appropriate
harmonic coordinates. In carrying out this program we prove moment and
conductance estimates for MGW trees, which may be of independent interest. In
addition, we extend our construction of the reversing measure to a biased
random walk with random environment (RWRE) on MGW trees, again at a critical
value of the bias. We compare this result against a transience-recurrence
criterion for the RWRE generalizing a result of Faraud (2011) for Galton-Watson
trees.
ELECTRONIC JOURNAL OF PROBABILITY 11/2010; 17. DOI:10.1214/EJP.v17-2294 · 0.77 Impact Factor