15 research outputs found
A Log-Sobolev Inequality for the Multislice, with Applications
Let kappa in N_+^l satisfy kappa_1 + *s + kappa_l = n, and let U_kappa denote the multislice of all strings u in [l]^n having exactly kappa_i coordinates equal to i, for all i in [l]. Consider the Markov chain on U_kappa where a step is a random transposition of two coordinates of u. We show that the log-Sobolev constant rho_kappa for the chain satisfies rho_kappa^{-1} <= n * sum_{i=1}^l 1/2 log_2(4n/kappa_i), which is sharp up to constants whenever l is constant. From this, we derive some consequences for small-set expansion and isoperimetry in the multislice, including a KKL Theorem, a Kruskal - Katona Theorem for the multislice, a Friedgut Junta Theorem, and a Nisan - Szegedy Theorem
Equivalent definitions for (degree one) Cameron-Liebler classes of generators in finite classical polar spaces
In this article, we study degree one Cameron-Liebler sets of generators in
all finite classical polar spaces, which is a particular type of a
Cameron-Liebler set of generators in this polar space, [9]. These degree one
Cameron-Liebler sets are defined similar to the Boolean degree one functions,
[15]. We summarize the equivalent definitions for these sets and give a
classification result for the degree one Cameron-Liebler sets in the polar
spaces W(5,q) and Q(6,q)
Boolean Function Analysis on High-Dimensional Expanders
We initiate the study of Boolean function analysis on high-dimensional expanders. We describe an analog of the Fourier expansion and of the Fourier levels on simplicial complexes, and generalize the FKN theorem to high-dimensional expanders.
Our results demonstrate that a high-dimensional expanding complex X can sometimes serve as a sparse model for the Boolean slice or hypercube, and quite possibly additional results from Boolean function analysis can be carried over to this sparse model. Therefore, this model can be viewed as a derandomization of the Boolean slice, containing |X(k)|=O(n) points in comparison to binom{n}{k+1} points in the (k+1)-slice (which consists of all n-bit strings with exactly k+1 ones)
Boolean functions on high-dimensional expanders
We initiate the study of Boolean function analysis on high-dimensional
expanders. We give a random-walk based definition of high-dimensional
expansion, which coincides with the earlier definition in terms of two-sided
link expanders. Using this definition, we describe an analog of the Fourier
expansion and the Fourier levels of the Boolean hypercube for simplicial
complexes. Our analog is a decomposition into approximate eigenspaces of random
walks associated with the simplicial complexes. Our random-walk definition and
the decomposition have the additional advantage that they extend to the more
general setting of posets, encompassing both high-dimensional expanders and the
Grassmann poset, which appears in recent work on the unique games conjecture.
We then use this decomposition to extend the Friedgut-Kalai-Naor theorem to
high-dimensional expanders. Our results demonstrate that a constant-degree
high-dimensional expander can sometimes serve as a sparse model for the Boolean
slice or hypercube, and quite possibly additional results from Boolean function
analysis can be carried over to this sparse model. Therefore, this model can be
viewed as a derandomization of the Boolean slice, containing only
points in contrast to points in the -slice
(which consists of all -bit strings with exactly ones).Comment: 48 pages, Extended version of the prior submission, with more details
of expanding posets (eposets
Boolean degree 1 functions on some classical association schemes
We investigate Boolean degree 1 functions for several classical association
schemes, including Johnson graphs, Grassmann graphs, graphs from polar spaces,
and bilinear forms graphs, as well as some other domains such as multislices
(Young subgroups of the symmetric group). In some settings, Boolean degree 1
functions are also known as \textit{completely regular strength 0 codes of
covering radius 1}, \textit{Cameron--Liebler line classes}, and \textit{tight
sets}.
We classify all Boolean degree functions on the multislice. On the
Grassmann scheme we show that all Boolean degree functions are
trivial for , and , and that
for general , the problem can be reduced to classifying all Boolean degree
functions on . We also consider polar spaces and the bilinear
forms graphs, giving evidence that all Boolean degree functions are trivial
for appropriate choices of the parameters.Comment: 22 pages; accepted by JCTA; corrected Conjecture 6.