55
55

Sep 20, 2013
09/13

by
Dmitry Panchenko

texts

#
eye 55

#
favorite 0

#
comment 0

The inequality of Vapnik and Chervonenkis controls the expectation of the function by its sample average uniformly over a VC-major class of functions taking into account the size of the expectation. Using Talagrand's kernel method we prove a similar result for the classes of functions for which Dudley's uniform entropy integral or bracketing entropy integral is finite.

Source: http://arxiv.org/abs/math/0405342v1

52
52

Sep 24, 2013
09/13

by
Dmitry Panchenko

texts

#
eye 52

#
favorite 0

#
comment 0

We conjecture that the Parisi functional in the SK model is convex in the functional order parameter and prove a partial result that shows the convexity along one-sided directions. A consequence of this result is log-convexity of L_1 norm for a class or random variables.

Source: http://arxiv.org/abs/math/0412463v1

48
48

Sep 20, 2013
09/13

by
Dmitry Panchenko

texts

#
eye 48

#
favorite 0

#
comment 0

This paper has been withdrawn since a trivial proof of the result has been pointed out to the author.

Source: http://arxiv.org/abs/math/0405361v3

125
125

Nov 14, 2013
11/13

by
Dmitry Panchenko

texts

#
eye 125

#
favorite 0

#
comment 0

Discrete Random Variable: - de�ned by probability function (p.f.)

Topics: Maths, Statistics and Probability, Mathematics

Source: http://www.flooved.com/reader/1626

64
64

Sep 20, 2013
09/13

by
Dmitry Panchenko

texts

#
eye 64

#
favorite 0

#
comment 0

Using Talagrand's concentration inequality on the discrete cube {0,1}^m we show that given a real-valued function Z(x)on {0,1}^m that satisfies certain monotonicity conditions one can control the deviations of Z(x) above its median by a local Lipschitz norm of Z(x) at the point x. As one application, we give a simple proof of a nearly optimal deviation inequality for the number of k-cycles in a random graph.

Source: http://arxiv.org/abs/math/0405355v1

8
8.0

Jun 28, 2018
06/18

by
Dmitry Panchenko

texts

#
eye 8

#
favorite 0

#
comment 0

Using the synchronization mechanism developed in the previous work on the Potts spin glass model, arXiv:1512.00370, we obtain the analogue of the Parisi formula for the free energy in the mixed even $p$-spin models with vector spins, which include the Sherrington-Kirkpatrick model with vector spins interacting through their scalar product. As a special case, this also establishes the sharpness of Talagrand's upper bound for the free energy of multiple mixed $p$-spin systems coupled by...

Topics: Probability, Mathematics, Mathematical Physics

Source: http://arxiv.org/abs/1512.04441

94
94

Nov 14, 2013
11/13

by
Dmitry Panchenko

texts

#
eye 94

#
favorite 0

#
comment 0

Topics: Maths, Statistics and Probability, Mathematics

Source: http://www.flooved.com/reader/1647

95
95

Nov 14, 2013
11/13

by
Dmitry Panchenko

texts

#
eye 95

#
favorite 0

#
comment 0

Topic: Maths

Source: http://www.flooved.com/reader/1029

65
65

Sep 20, 2013
09/13

by
Dmitry Panchenko

texts

#
eye 65

#
favorite 0

#
comment 0

In this paper we consider a system of spins that consists of two configurations $\vsi^1,\vsi^2\in\Sigma_N=\{-1,+1\}^N$ with Gaussian Hamiltonians $H_N^1(\vsi^1)$ and $H_N^2(\vsi^2)$ correspondingly, and these configurations are coupled on the set where their overlap is fixed $\{R_{1,2}=N^{-1}\sum_{i=1}^N \sigma_i^1\sigma_i^2 = u_N\}.$ We prove the existence of the thermodynamic limit of the free energy of this system given that $\lim_{N\to\infty}u_N = u\in[-1,1]$ and give the analogue of the...

Source: http://arxiv.org/abs/math/0405359v1

58
58

Sep 19, 2013
09/13

by
Dmitry Panchenko

texts

#
eye 58

#
favorite 0

#
comment 0

Recently, Michel Talagrand computed the large deviations limit $\lim_{N\to\infty}(Na)^{-1}\log \e Z_N^a$ for the moments of the partition function $Z_N$ in the Sherrington-Kirkpatrick model for all real $a\geq 0.$ For $a\geq 1$ the limit is given by Guerra's inverse bound and this result extends the classical physicist's replica method that corresponds to integer $a.$ We give a new proof for $a\geq 1$ in the case of the pure $p$-spin SK model that provides a strong exponential control of the...

Source: http://arxiv.org/abs/math-ph/0701074v1

66
66

Sep 24, 2013
09/13

by
Dmitry Panchenko

texts

#
eye 66

#
favorite 0

#
comment 0

In this paper we prove that the support of a random measure on the unit ball of a separable Hilbert space that satisfies the Ghirlanda-Guerra identities must be ultrametric with probability one. This implies the Parisi ultrametricity conjecture in mean-field spin glass models, such as the Sherrington-Kirkpatrick and mixed $p$-spin models, for which Gibbs' measures are known to satisfy the Ghirlanda-Guerra identities in the thermodynamic limit.

Source: http://arxiv.org/abs/1112.1003v1

50
50

Sep 21, 2013
09/13

by
Dmitry Panchenko

texts

#
eye 50

#
favorite 0

#
comment 0

Gibbs' measures in the Sherrington-Kirkpatrick type models satisfy two asymptotic stability properties, the Aizenman-Contucci stochastic stability and the Ghirlanda-Guerra identities, which play a fundamental role in our current understanding of these models. In this paper we show that one can combine these two properties very naturally into one unified stability property.

Source: http://arxiv.org/abs/1106.3954v3

137
137

Nov 14, 2013
11/13

by
Dmitry Panchenko

texts

#
eye 137

#
favorite 0

#
comment 0

Topics: Maths, Analysis and Calculus, Statistics and Probability, Mathematics

Source: http://www.flooved.com/reader/1031

285
285

Nov 14, 2013
11/13

by
Dmitry Panchenko

texts

#
eye 285

#
favorite 1

#
comment 0

Topics: Maths, Analysis and Calculus, Statistics and Probability, Mathematics

Source: http://www.flooved.com/reader/1021

150
150

Nov 14, 2013
11/13

by
Dmitry Panchenko

texts

#
eye 150

#
favorite 0

#
comment 0

Transforms the outcome of an experiment into a number. De�nitions: Probability Space: (S, A, P) S - sample space, A - events, P - probability Random variable is a function on S with values in real numbers, X:S_R

Topics: Maths, Statistics and Probability, Mathematics

Source: http://www.flooved.com/reader/1646

97
97

Jul 20, 2013
07/13

by
Dmitry Panchenko

texts

#
eye 97

#
favorite 0

#
comment 0

In this paper we translate Talagrand's solution of the K-sat model at high temperature into the language of asymptotic Gibbs measures. Using exact cavity equations in the infinite volume limit allows us to remove many technicalities of the inductions on the system size, which clarifies the main ideas of the proof. This approach also yields a larger region of parameters where the system is in a pure state and, in particular, for small connectivity parameter we prove the replica symmetric formula...

Source: http://arxiv.org/abs/1304.6052v1

227
227

Nov 14, 2013
11/13

by
Dmitry Panchenko

texts

#
eye 227

#
favorite 0

#
comment 0

Topics: Maths, Analysis and Calculus, Statistics and Probability, Measure Theory, Probability,...

Source: http://www.flooved.com/reader/1033

7
7.0

Jun 30, 2018
06/18

by
Dmitry Panchenko

texts

#
eye 7

#
favorite 0

#
comment 0

We suggest a possible approach to proving the M\'ezard-Parisi formula for the free energy in the diluted spin glass models, such as diluted K-spin or random K-sat model at any positive temperature. In the main contribution of the paper, we show that a certain small modification of the Hamiltonian in any of these models forces all finite-RSB asymptotic Gibbs measures in the sense of the overlaps to satisfy the M\'ezard-Parisi ansatz for the distribution of spins. Unfortunately, what is still...

Topics: Probability, Mathematics, Mathematical Physics

Source: http://arxiv.org/abs/1406.4702

147
147

Nov 14, 2013
11/13

by
Dmitry Panchenko

texts

#
eye 147

#
favorite 0

#
comment 0

Topics: Maths, Statistics and Probability, Mathematics

Source: http://www.flooved.com/reader/1644

125
125

Nov 14, 2013
11/13

by
Dmitry Panchenko

texts

#
eye 125

#
favorite 1

#
comment 0

Given the joint distribution of (X, Y), the individual distributions of X, Y are marginal distributions.

Topics: Maths, Statistics and Probability, Mathematics

Source: http://www.flooved.com/reader/1636

174
174

Nov 14, 2013
11/13

by
Dmitry Panchenko

texts

#
eye 174

#
favorite 0

#
comment 0

Covariance of X and Y is de�ned as: ... Positive when both high or low in deviation

Topics: Maths, Statistics and Probability, Mathematics

Source: http://www.flooved.com/reader/1625

117
117

Nov 14, 2013
11/13

by
Dmitry Panchenko

texts

#
eye 117

#
favorite 0

#
comment 0

In this section we will make several connections between the Kantorovich-Rubinstein theorem and other classical objects. Let us start with the following classical inequality.

Topics: Maths, Analysis and Calculus, Statistics and Probability, Mathematics

Source: http://www.flooved.com/reader/1038

125
125

Nov 14, 2013
11/13

by
Dmitry Panchenko

texts

#
eye 125

#
favorite 0

#
comment 0

Topics: Maths, Analysis and Calculus, Statistics and Probability, Mathematics

Source: http://www.flooved.com/reader/1020

92
92

Nov 14, 2013
11/13

by
Dmitry Panchenko

texts

#
eye 92

#
favorite 0

#
comment 0

Topics: Maths, Statistics and Probability, Mathematics

Source: http://www.flooved.com/reader/1635

128
128

Nov 14, 2013
11/13

by
Dmitry Panchenko

texts

#
eye 128

#
favorite 0

#
comment 0

t-test X1, ..., Xn - a random sample from N(�, _^2)

Topics: Maths, Statistics and Probability, Mathematics

Source: http://www.flooved.com/reader/1652

141
141

Nov 14, 2013
11/13

by
Dmitry Panchenko

texts

#
eye 141

#
favorite 0

#
comment 0

Topics: Maths, Analysis and Calculus, Statistics and Probability, Measure Theory, Probability, ?-Algebras,...

Source: http://www.flooved.com/reader/1035

144
144

Nov 14, 2013
11/13

by
Dmitry Panchenko

texts

#
eye 144

#
favorite 0

#
comment 0

Topics: Maths, Statistics and Probability, Mathematics

Source: http://www.flooved.com/reader/1620

129
129

Jul 19, 2013
07/13

by
Dmitry Panchenko

texts

#
eye 129

#
favorite 0

#
comment 0

In many spin glass models, due to the symmetry among sites, any limiting joint distribution of spins under the annealed Gibbs measure admits the Aldous-Hoover representation encoded by a function $\sigma:[0,1]^4\to\{-1,+1\}$, and one can think of this function as a generic functional order parameter of the model. In a class of diluted models, and in the Sherrington-Kirkpatrick model, we introduce novel perturbations of the Hamiltonian that yield certain invariance and self-consistency equations...

Source: http://arxiv.org/abs/1005.2720v2

46
46

Sep 20, 2013
09/13

by
Dmitry Panchenko

texts

#
eye 46

#
favorite 0

#
comment 0

We show that, under the conditions known to imply the validity of the Parisi formula, if the generic Sherrington-Kirkpatrick Hamiltonian contains a $p$-spin term then the Ghirlanda-Guerra identities for the $p$th power of the overlap hold in a strong sense without averaging. This implies strong version of the extended Ghirlanda-Guerra identities for mixed $p$-spin models than contain terms for all even $p\geq 2$ and $p=1.$

Source: http://arxiv.org/abs/1002.2190v1

135
135

Nov 14, 2013
11/13

by
Dmitry Panchenko

texts

#
eye 135

#
favorite 1

#
comment 0

Bayes Estimators: used when intuitive model can be used in describing the data.

Topics: Maths, Statistics and Probability, Mathematics

Source: http://www.flooved.com/reader/1627

96
96

Nov 14, 2013
11/13

by
Dmitry Panchenko

texts

#
eye 96

#
favorite 0

#
comment 0

Topic: Maths

Source: http://www.flooved.com/reader/1034

13
13

Jun 26, 2018
06/18

by
Dmitry Panchenko

texts

#
eye 13

#
favorite 0

#
comment 0

We prove chaos in temperature for even $p$-spin models which include sufficiently many $p$-spin interaction terms. Our approach is based on a new invariance property for coupled asymptotic Gibbs measures, similar in spirit to the invariance property that appeared in the proof of ultrametricity in arXiv:1112.1003, used in combination with Talagrand's analogue of Guerra's replica symmetry breaking bound for coupled systems.

Topics: Mathematics, Probability, Mathematical Physics

Source: http://arxiv.org/abs/1502.03801

60
60

Sep 21, 2013
09/13

by
Dmitry Panchenko

texts

#
eye 60

#
favorite 0

#
comment 0

In this paper we obtain a new family of identities for random measures on the unit ball of a separable Hilbert space which arise as the asymptotic analogues of the Gibbs measures in the Sherrington-Kirkpatrick and $p$-spin models and which are known to satisfy the Ghirlanda-Guerra identities. We give several applications of the new identities to structural results for such measures.

Source: http://arxiv.org/abs/1108.0379v2

143
143

Nov 14, 2013
11/13

by
Dmitry Panchenko

texts

#
eye 143

#
favorite 0

#
comment 0

Topics: Maths, Analysis and Calculus, Statistics and Probability, Mathematics

Source: http://www.flooved.com/reader/1041

402
402

Nov 14, 2013
11/13

by
Dmitry Panchenko

texts

#
eye 402

#
favorite 1

#
comment 0

De�nition: Conditional probability of Event A given Event B: ...

Topics: Maths, Statistics and Probability, Mathematics

Source: http://www.flooved.com/reader/1637

142
142

Nov 14, 2013
11/13

by
Dmitry Panchenko

texts

#
eye 142

#
favorite 0

#
comment 0

_(_), parameter _ > 0, random variable takes values: {0, 1, 2, ...} p.f. ... Moment generating function:

Topics: Maths, Statistics and Probability, Probability, Statistics, Random Variables, Useful Distributions,...

Source: http://www.flooved.com/reader/1642

6
6.0

Jun 30, 2018
06/18

by
Dmitry Panchenko

texts

#
eye 6

#
favorite 0

#
comment 0

This is a review paper for the "Current Developments in Mathematics 2014" conference.

Topics: Probability, Mathematics, Mathematical Physics

Source: http://arxiv.org/abs/1412.0170

58
58

Sep 18, 2013
09/13

by
Dmitry Panchenko

texts

#
eye 58

#
favorite 0

#
comment 0

The goal of this paper is to review some of the main ideas that emerged from the attempts to confirm mathematically the predictions of the celebrated Parisi ansatz in the Sherrington-Kirkpatrick model. We try to focus on the big picture while sketching the proofs of only a few selected results, but an interested reader can find most of the missing details in [31] and [44].

Source: http://arxiv.org/abs/1211.1094v1

68
68

Sep 20, 2013
09/13

by
Dmitry Panchenko

texts

#
eye 68

#
favorite 0

#
comment 0

In this paper we revisit Talagrand's proof of concentration inequality for empirical processes. We give a different shorter proof of the main technical lemma that garantees the existence of a certain kernel. Our proof provides the almost optimal value of the constant involved in the statement of this lemma.

Source: http://arxiv.org/abs/math/0405341v1

101
101

Nov 14, 2013
11/13

by
Dmitry Panchenko

texts

#
eye 101

#
favorite 0

#
comment 0

Topic: Maths

Source: http://www.flooved.com/reader/1030

124
124

Nov 14, 2013
11/13

by
Dmitry Panchenko

texts

#
eye 124

#
favorite 0

#
comment 0

Topics: Maths, Analysis and Calculus, Statistics and Probability, Mathematics

Source: http://www.flooved.com/reader/1025

113
113

Nov 14, 2013
11/13

by
Dmitry Panchenko

texts

#
eye 113

#
favorite 0

#
comment 0

Topics: Maths, Statistics and Probability, Statistics, Statistical Tables, Contingency Tables, Mathematics

Source: http://www.flooved.com/reader/1624

106
106

Nov 14, 2013
11/13

by
Dmitry Panchenko

texts

#
eye 106

#
favorite 0

#
comment 0

Topics: Maths, Statistics and Probability, Mathematics

Source: http://www.flooved.com/reader/1632

196
196

Nov 14, 2013
11/13

by
Dmitry Panchenko

texts

#
eye 196

#
favorite 0

#
comment 0

Topics: Maths, Statistics and Probability, Probability, Statistics, Bayesian Statistics, Bayes Estimates,...

Source: http://www.flooved.com/reader/1618

117
117

Nov 14, 2013
11/13

by
Dmitry Panchenko

texts

#
eye 117

#
favorite 0

#
comment 0

In this section we will begin the discussion of weak convergence of distributions on metric spaces. Let (S, d) be a metric space with a metric d. Consider a measurable space (S, B) with Borel _-algebra B generated by open sets and let (Pn)n�1 and P be probability distributions on B

Topics: Maths, Analysis and Calculus, Statistics and Probability, Mathematics

Source: http://www.flooved.com/reader/1023

276
276

Nov 14, 2013
11/13

by
Dmitry Panchenko

texts

#
eye 276

#
favorite 0

#
comment 0

Topics: Maths, Analysis and Calculus, Statistics and Probability, Mathematics

Source: http://www.flooved.com/reader/1039

5
5.0

Jun 28, 2018
06/18

by
Dmitry Panchenko

texts

#
eye 5

#
favorite 0

#
comment 0

We study the Potts spin glass model, which generalizes the Sherrington-Kirkpatrick model to the case when spins take more than two values but their interactions are counted only if the spins are equal. We obtain the analogue of the Parisi variational formula for the free energy, with the order parameter now given by a monotone path in the set of positive-semidefinite matrices. The main idea of the paper is a novel synchronization mechanism for blocks of overlaps. This mechanism can be used to...

Topics: Probability, Mathematics, Mathematical Physics

Source: http://arxiv.org/abs/1512.00370

112
112

Nov 14, 2013
11/13

by
Dmitry Panchenko

texts

#
eye 112

#
favorite 0

#
comment 0

Topic: Maths

Source: http://www.flooved.com/reader/1024

105
105

Nov 14, 2013
11/13

by
Dmitry Panchenko

texts

#
eye 105

#
favorite 0

#
comment 0

Topics: Maths, Statistics and Probability, Mathematics

Source: http://www.flooved.com/reader/1630

142
142

Nov 14, 2013
11/13

by
Dmitry Panchenko

texts

#
eye 142

#
favorite 0

#
comment 0

Topics: Maths, Analysis and Calculus, Statistics and Probability, Mathematics

Source: http://www.flooved.com/reader/1028