Subgaussian concentration and rates of convergence in directed polymers

Abstract

We consider directed random polymers in $(d+1)$ dimensions with nearly gamma i.i.d. disorder. We study the partition function $Z_{N,\omega}$ and establish exponential concentration of $\log Z_{N,\omega}$ about its mean on the subgaussian scale $\sqrt{N/\log N}$ . This is used to show that $\mathbb{E}[ \log Z_{N,\omega}]$ differs from $N$ times the free energy by an amount which is also subgaussian (i.e. $o(\sqrt{N})$), specifically $O(\sqrt{\frac{N}{\log N}}\log \log N)$.Comment: Minor changes. Appears in Electronic Journal of Probability, 18, 2013, no.

Rate of convergence in first-passage percolation under low moments

“…(1.7) may be used in Alexander's method to obtain our Proposition 1.1, but with a worse moment condition. One may ask if it can be used to prove a sub-gaussian bound Eτ (0, x) ≤ µ(x) + o( x 1 ), as in the case of directed polymers [4], but no such theorem exists in an undirected model. The main complication arises from the extra logarithmic factor coming from Alexander's method, which negates the gain in the scale from (1.7).…”

Localization for Random Walks Among Random Obstacles in a Single Euclidean Ball

“…Rate of convergence for sub-additive functionals has received much attention in the past. See [2,32,3,4,10] for progresses on bounds for rate of convergence for sub-additive functionals with prominent application in first-passage percolation. In particular, a general theory was given in [3] via the ingenious convex hull approximation property which applies to several processes on lattices including first-passage percolation.…”

“…[1]), which we denote as C K (∞). Then the following holds with probability at least 1 − e −(log |x−y|) 4…”

“…We remark that we do not require either 0 or N x is in C(∞) here. We verify the condition in Lemma 4.24 for M = n 1/3 and a = 4 as a consequence of the following lemma, and then prove Lemma 4.22. with α i ≥ 0, d+1 i=1 α i ∈ [1,4] and y i ∈ Q x/q . Let x * = d+1 i=1 qα i y i .…”

Sublinear variance in Euclidean first-passage percolation

“…One would like to use these tools to prove, as has been done in directed polymers [AZ13], that the error term o(n) above is actually o( √ n). To do this, one needs to be able to show (1) using ψ(n) = n/ log n. This would effectively bound the random fluctuation term by o( √ n).…”