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• 12:37, 14 January 2026 Lptmswikids talk contribs created page More general REM and systems in Finite dimensions (Created page with "In random energy models with i.i.d. random variables, the distribution <math>p(E)</math> determines the dependence of <math>a_M</math> and <math>b_M</math> on ''M'', and consequently their scaling with ''N'', the number of degrees of freedom. It is insightful to consider a more general case where an exponent <math>\omega</math> describes the fluctuations of the ground state energy: <center> <math>\overline{\left(E_{\min} - \overline{E_{\min}}\right)^2} \sim b_M^2 \propto...")
• 12:37, 14 January 2026 Lptmswikids talk contribs created page Behavior in Different Phases (Created page with "* '''High-Temperature Phase (<math> T > T_f= b_M = 1/\sqrt{2 \log2}</math>):''' : In this regime, the weight of the excited states diverges. This indicates that the ground state is not deep enough to render the system glassy. * '''Low-Temperature Phase (<math> T < T_f= b_M = 1/\sqrt{2 \log2}</math>):''' : In this regime, the integral is finite: <center> <math> \int_0^\infty dx \, e^{ (1/b_M-\beta) x}/b_M = \frac{1}{\beta b_M-1} = \frac{T}{T_f - T} </math> </c...")
• 12:36, 14 January 2026 Lptmswikids talk contribs created page Phase Transition in the Random Energy Model (Created page with "The Random Energy Model (REM) exhibits two distinct phases: * '''High-Temperature Phase:''' : At high temperatures, the system is in a paramagnetic phase where the entropy is extensive, and the occupation probability of a configuration is approximately <math>\sim 1/M</math>. * '''Low-Temperature Phase:''' : Below a critical freezing temperature <math>T_f</math>, the system transitions into a glassy phase. In this phase, the entropy becomes subextensive (i.e., the...")
• 12:36, 14 January 2026 Lptmswikids talk contribs created page Back to the REM (Created page with "Returning now to the Random Energy Model (REM), recall that for each realization of disorder, we obtain <math>M=2^N</math> Gaussian random variables with zero mean and variance <math>\sigma^2_M = \frac{\log M}{\log 2} = N</math>. The minimum energy is a random variable belonging to the Gumbel universality class. From the results for <math>a_M </math> and <math>b_M</math> derived in the previous section, we have: <center> <math>E_{\min} =a_M +b_M z = - \sqrt{2 \log 2}\;...")
• 12:36, 14 January 2026 Lptmswikids talk contribs created page Density above the minimum (Created page with "'''Definition of <math> n(x) </math>:''' Given a realization, <math> n(x) </math> is defined as the number of random variables above the minimum <math>E_{\min} </math> such that their value is smaller than <math>E_{\min} +x</math>. This quantity is a random variable, and we are interested in its average value: <center><math> \overline{n(x)} = \sum_k k \, \text{Prob}[n(x) = k] </math></center> The key relation for this quantity is: <center><math> \text{Prob}[n(x) = k]...")
• 12:35, 14 January 2026 Lptmswikids talk contribs created page The Scaling Form in the Large ''M'' Limit (Created page with "In the spirit of the central limit theorem, we look for a scaling form: <center> <math>E_{\min} = a_M + b_M z</math> </center> The constants <math>a_M</math> and <math>b_M</math> absorb the dependence on <math>M</math>, while the random variable <math>z</math> is distributed according to a probability distribution <math>P(z)</math> that does not depend on <math>M</math>. In the Gaussian case, we start from the third relation introduced earlier and expand <math>A(E)</mat...")
• 12:35, 14 January 2026 Lptmswikids talk contribs created page A Concrete Example: The Gaussian Case (Created page with "To understand how a universal scaling form emerges, let us analyze in detail the case of a Gaussian distribution with zero mean and variance <math>\sigma^2</math>. Using integration by parts, we can write : <center> <math>P^<(E) = \int_{-\infty}^E \frac{dx}{\sqrt{2 \pi \sigma^2}} \, e^{-\frac{x^2}{2 \sigma^2}} = \frac{1}{2\sqrt{\pi}} \int_{\frac{E^2}{2 \sigma^2}}^{\infty} \frac{dt}{\sqrt{t}} e^{-t}= \frac{\sigma}{\sqrt{2 \pi} |E|} e^{-\frac{E^2}{2 \sigma^2}} - \frac{1}{4...")
• 12:34, 14 January 2026 Lptmswikids talk contribs created page Detour: Extreme Value Statistics (Created page with "Consider the <math>M</math> energies <math>E_1, \dots, E_M</math> as independent and identically distributed (i.i.d.) random variables drawn from a distribution <math>p(E)</math>. It is useful to introduce the cumulative probability of finding an energy smaller than ''E'': <center> <math>P^<(E) = \int_{-\infty}^E dx \, p(x)</math> </center> The complementary probability of finding an energy larger than ''E'' is: <center> <math>P^>(E) = \int_E^{+\infty} dx \, p(x) = 1 -...")
• 12:34, 14 January 2026 Lptmswikids talk contribs created page Random energy model (Created page with "The solution of the Sherrington-Kirkpatrick (SK) model is challenging. To make progress, we first study the Random Energy Model (REM), introduced by B. Derrida. This model simplifies the problem by neglecting correlations between the <math>M=2^N</math> configurations and assuming that the energies <math>E_{\alpha}</math> are independent and identically distributed (i.i.d.) random variables. Here, "independent" means that the energy of one configuration does not influence...")
• 12:29, 14 January 2026 Lptmswikids talk contribs created page The SK model (Created page with "== The SK model == Sherrington and Kirkpatrik considered the fully connected version of the model with Gaussian couplings: <center> <math> E= - \sum_{i,j} \frac{J_{ij}}{ \sqrt{N}} \sigma_i \sigma_j </math></center> At the inverse temperature <math> \beta </math>, the partion function of the model is <center> <math> Z= \sum_{\alpha=1}^{2^N} z_{\alpha}, \quad \text{with}\; z_{\alpha}= e^{-\beta E_\alpha} </math></center> Here <math> E_\alpha </math> is the energy asso...")
• 00:00, 14 January 2026 Lptmswikids talk contribs created page Test-qin (Created page with "== Back to REM == In the REM, the variance of the energies scales with the system size as <center><math>\sigma_M^2 = \frac{\log M}{\log 2} = N.</math></center> As a consequence, the minimum energy takes the form <center><math> E_{\min} = a_M + b_M z = - \sqrt{2 \log 2}\, N + \frac{1}{2}\, \frac{\log (4 \pi N \log 2)}{\sqrt{2 \log 2}} + \frac{z}{\sqrt{2 \log 2}}, </math></center> where <math>z</math> is a Gumbel-distributed random variable. '''Key observations:''' *...")
• 07:19, 13 January 2026 Rosso talk contribs created page L1 ICTS (Created page with "<strong>Goal: </strong> Spin glass transition. From the experiments with the anomaly on the magnetic susceptibility to order parameter of the transition. We will discuss the arguments linked to extreme value statistics = Spin glass Transition = Spin glass behavior was first observed in experiments with non-magnetic metals (such as Cu, Fe, Au, etc.) doped with a small percentage of magnetic impurities, typically Mn. At low doping levels, the magnetic moments of Mn...")
• 09:11, 9 September 2025 Rosso talk contribs created page TBan-IV (Created page with "== Bienaymé Galton Watson process== A time <math> t=0 </math> appears as infected individual which dies with a rate <math> a </math> and branches with a rate <math> b </math>. On average, each infection generates in average <math> R_0 = b/a </math> new ones. Real epidemics corresponds to <math> R_0>1 </math>. At time <math> t </math>, the infected population is <math> n(t) </math>, while the total infected population is <center> <math> N(t) = \int_0^t n(t') d t'...")
• 21:59, 8 September 2025 Rosso talk contribs created page LBan-V (Created page with "<Strong> Goal: </Strong> We solve the mean field version of the cellular automaton, derive its avalanche statistics and make a connection with the Bienaymé-Galton-Watson process used to describe an epidemic outbreak. = Fully connected (mean field) model for the cellular automaton= Let's study the mean field version of the cellular automata introduced in the previous lecture. We introduce two approximations: * Replace the Laplacian, which is short range, with a mea...")
• 18:31, 31 August 2025 Rosso talk contribs created page LBan-IV (Created page with "= Pinning and Depinning of a Disordered Material = In earlier lectures, we discussed how disordered systems can become trapped in deep energy states, forming a glass. Today, we will examine how such systems can also be ''pinned'' and resist external deformation. This behavior arises because disorder creates a complex energy landscape with numerous features, including minima (of varying depth), maxima, and saddle points. When an external force is applied, it tilts t...")
• 15:50, 31 August 2025 Rosso talk contribs created page TBan-III (Created page with "= Exercise 2: Edwards-Wilkinson Interface with Stationary Initial Condition (7.5 points) = Consider an Edwards-Wilkinson interface in 1+1 dimensions, at temperature <math>T</math>, and of length <math>L</math> with periodic boundary conditions: <center><math> \frac{\partial h(x,t)}{\partial t} = \nu \nabla^2 h(x,t) + \eta(x,t) </math></center> where <math>\eta(x,t)</math> is a Gaussian white noise with zero mean and variance: <center><math> \langle \eta(x,t) \eta(x',...")
• 20:26, 30 August 2025 Rosso talk contribs created page TBan-II (Created page with "==Dijkstra Algorithm and transfer matrix== thumb|left|Sketch of the discrete Directed Polymer model. At each time the polymer grows either one step left either one step right. A random energy <math> V(\tau,x)</math> is associated at each node and the total energy is simply <math> E[x(\tau)] =\sum_{\tau=0}^t V(\tau,x)</math>. We introduce a lattice model for the directed polymer (see figure). In a companion notebook we provide the implementa...")
• 19:10, 27 August 2025 Rosso talk contribs created page LBan-III (Created page with "= Goal = The goal of this lecture is to present the final lecture on KPZ and directed polymers in finite dimension. We show that for <math>d>2</math>, a "glass transition" occurs. = KPZ: from 1D to the Cayley tree = Much is known about KPZ, but several aspects remain elusive: In <math>d=1</math>, we have <math>\theta=1/3</math> and a glassy regime present at all temperatures. The stationary solution of the KPZ equation describes, at long times, the fluctuations of qu...")
• 21:06, 6 August 2025 Rosso talk contribs created page LBan-II (Created page with "=Edwards Wilkinson: an interface with thermal fluctuations: = Consider domain wall <math> h(r,t)</math> fluctuating at equilibrium at the temparature <math> T</math>. Here <math> t</math> is time, <math> r </math> defines the d-dimensional coordinate of the interface and <math> h</math> is the scalar height field. Hence, the domain wall separating two phases in a film has <math> d=1, r \in \cal{R}</math>, in a solid instead <math> d=2, r \in \cal{R}^2</math>. Two...")
• 16:23, 6 August 2025 Rosso talk contribs created page TBan-I (Created page with "ciao")
• 16:18, 31 July 2025 Rosso talk contribs created page LBan-1 (Created page with "aaa")
• 12:25, 31 July 2025 Rosso talk contribs created page ICTS (Created page with "== Course description == This course deals with systems in which the presence of impurities or amorphous structures (in other words, of disorder) influences radically the physics, generating novel phenomena. These phenomena involve the properties of the system at equilibrium (freezing transitions, glassy phase and glassy system), as well as their dynamical evolution out-of-equilibrium (pinning, avalanches). * The simplest spin-glass: solution of the Random Energy Mode...")
• 12:54, 24 March 2025 Ros talk contribs created page File:2025 localization notes.pdf
• 12:54, 24 March 2025 Ros talk contribs uploaded File:2025 localization notes.pdf
• 15:43, 23 March 2025 Ros talk contribs created page File:2025 P9 solutions.pdf
• 15:43, 23 March 2025 Ros talk contribs uploaded File:2025 P9 solutions.pdf
• 15:42, 23 March 2025 Ros talk contribs deleted page File:2024 TD9 solutions.pdf
• 16:02, 19 March 2025 Ros talk contribs created page File:2025 Self energies, decay rates and all that.pdf
• 16:02, 19 March 2025 Ros talk contribs uploaded File:2025 Self energies, decay rates and all that.pdf
• 16:02, 19 March 2025 Ros talk contribs deleted page File:Self energies, decay rates and all that.pdf
• 15:58, 19 March 2025 Ros talk contribs created page File:Self energies, decay rates and all that.pdf
• 15:58, 19 March 2025 Ros talk contribs uploaded File:Self energies, decay rates and all that.pdf
• 15:58, 19 March 2025 Ros talk contribs deleted page File:Self energies, decay rates and all that.pdf
• 13:32, 19 March 2025 Ros talk contribs created page File:Self energies, decay rates and all that.pdf
• 13:32, 19 March 2025 Ros talk contribs uploaded File:Self energies, decay rates and all that.pdf
• 20:24, 16 March 2025 Ros talk contribs created page File:2025 P8 solutions.pdf
• 20:24, 16 March 2025 Ros talk contribs uploaded File:2025 P8 solutions.pdf
• 20:23, 16 March 2025 Ros talk contribs deleted page File:2024 TD8 solutions.pdf
• 11:41, 12 March 2025 Ros talk contribs created page File:2025 P7 solutions .pdf
• 11:41, 12 March 2025 Ros talk contribs uploaded File:2025 P7 solutions .pdf
• 11:40, 12 March 2025 Ros talk contribs deleted page File:2024 TD7 solutions.pdf
• 16:21, 2 March 2025 Ros talk contribs created page File:2025 P666 solutions .pdf
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• 16:19, 2 March 2025 Ros talk contribs deleted page File:2025 P6 solutions .pdf
• 16:16, 2 March 2025 Ros talk contribs created page File:2025 P6 solutions .pdf
• 16:16, 2 March 2025 Ros talk contribs uploaded File:2025 P6 solutions .pdf
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• 16:13, 2 March 2025 Ros talk contribs created page File:2025 P6 solutions .pdf
• 16:13, 2 March 2025 Ros talk contribs uploaded File:2025 P6 solutions .pdf
• 16:13, 2 March 2025 Ros talk contribs deleted page File:2025 P6 solutions .pdf