Lecture 6 . The Cluster Expansion Lemma

نویسنده

  • Erik Bates
چکیده

We have seen the Lovász Local Lemma and its stronger variant, Shearer’s Lemma, which is unfortunately quite unwieldy in applications. Quite recently, researchers in mathematical physics discovered an intermediate form of the lemma, which seems to give results close to Shearer’s Lemma but it is much more easily applicable. First let us review a connection between Shearer’s Lemma and statistical physics which inspired this development.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Lecture 7 . The Cluster Expansion Lemma

We have seen the Lovász Local Lemma and its stronger variant, Shearer’s Lemma, which is unfortunately quite unwieldy in applications. Quite recently, researchers in mathematical physics discovered an intermediate form of the lemma, which seems to give results close to Shearer’s Lemma but it is much more easily applicable. First let us review a connection between Shearer’s Lemma and statistical ...

متن کامل

Axiomatic Set Theory

Lecture 1, 07.03.: We made a review of the material covered in Chapter I of [3], up to Theorem I.9.11 (Transfinite Recursion on Well-founded Relations). Lecture 2, 14.03.: We discussed the notion of a rank, as well as the Mostowski collapsing function material corresponding to Section 9 of [3]. Lecture 3, 04.04.: We discussed hereditarily transitive sets, the DownwardLöwenheim-Skolem-Tarksi The...

متن کامل

2005 ) Lecture 3 : Expander Graphs and PCP Theorem Proof Overview

Recall in last lecture that we defined a (n, d, λ)-expander to be a d-regular n-vertex undirected graph with second eigenvalue λ. We also defined the edge expansion of a graph G with vertex set V to be φ(G) = min S⊂V |S|≤n/2 |E(S, S)| |S| , where E(S, S) is the set of edges between a vertex set S and its complement. The following lemma shows that the eigenvalue formulation of an expander is ess...

متن کامل

236779: Foundations of Algorithms for Massive Datasets Lecture 4 the Johnson-lindenstrauss Lemma

The Johnson-Lindenstrauss lemma and its proof This lecture aims to prove the Johnson–Lindenstrauss lemma. Since the lemma is proved easily with another interesting lemma, a part of this lecture is focused on the proof of this second lemma. At the end, the optimality of the Johnson–Lindenstrauss lemma is discussed. Lemma 1 (Johnson-Lindenstrauss). Given the initial space X ⊆ R n s.t. |X| = N , <...

متن کامل

Lecture 4 : Leftover Hash Lemma and One Way Functions

We recall some definitions and a claim proved in our previous lecture. These will be required to finish the proof for the Leftover Hash Lemma. Definition 1 H∞(X) = − log (maxx Pr [X = x]) ♦ Definition 2 A function Ext : U × S → V is a (k, ε) extractor if for all random variables X with H∞(X) ≥ k, we have: SD [(S,Ext(X,S)) , (S, V )] ≤ ε where S is uniformly distributed over S and V is uniformly...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2018