Using a geometric argument, we show that under a reasonable continuity condition, the Clarke subdifferential of a semi-algebraic (or more generally stratifiable) directionally Lipschitzian function admits a simple form: the normal cone to the domain and limits of gradients generate the entire Clarke subdifferential. The characterization formula we obtain unifies various apparently disparate results that have appeared in the literature. Our techniques also yield a simplified proof that closed semialgebraic functions on $R^n$ have a limiting subdifferential graph of uniform local dimension $n$.
We consider the Bernoulli bond percolation process $mathbb{P}_{p,p}$ on the nearest-neighbor edges of $mathbb{Z}^d$, which are open independently with probability $p<p_c$, except for those lying on the first coordinate axis, for which this probability is $p$. Define [xi_{p,p}:=-lim_{ntoinfty}n^{-1}log mathbb{P}_{p,p}(0leftrightarrow nmathbf {e}_1)] and $xi_p:=xi_{p,p}$. We show that there exists $p_c=p_c(p,d)$ such that $xi_{p,p}=xi_p$ if $p<p_c$ and $xi_{p,p}<xi_p$ if $p>p_c$. Moreover, $p_c(p,2)=p_c(p,3)=p$, and $p_c(p,d)>p$ for $dgeq 4$. We also analyze the behavior of $xi_p-xi_{p,p}$ as $pdownarrow p_c$ in dimensions $d=2,3$. Finally, we prove that when $p>p_c$, the following purely exponential asymptotics holds: [mathbb {P}_{p,p}(0leftrightarrow nmathbf {e}_1)=psi_de^{-xi_{p,p}n}bigl(1+o(1)bigr)] for some constant $psi_d=psi_d(p,p)$, uniformly for large values of $n$. This work gives the first results on the rigorous analysis of pinning-type problems, that go beyond the effective models and dont rely on exact computations.
In this paper we study the metastable behavior of one of the simplest disordered spin system, the random field Curie-Weiss model. We will show how the potential theoretic approach can be used to prove sharp estimates on capacities and metastable exit times also in the case when the distribution of the random field is continuous. Previous work was restricted to the case when the random field takes only finitely many values, which allowed the reduction to a finite dimensional problem using lumping techniques. Here we produce the first genuine sharp estimates in a context where entropy is important.
