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• 17:44, 14 January 2026 Lptmswikids talk contribs deleted page L-1-new (content was: "<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 * Random energy model * Detour: Extreme Value Statistics * A Concrete Example: The Gaussian Case * The Scaling Form in the Large ''M'...", and the only contributor was "Lptmswikids" (talk))
• 17:36, 14 January 2026 Lptmswikids talk contribs deleted page The Scaling Form in the Large ''M'' Limit (content was: "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</...", and the only contributor was "Lptmswikids" (talk))
• 17:35, 14 January 2026 Lptmswikids talk contribs deleted page The SK model (content was: "== 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 \te...", and the only contributor was "Lptmswikids" (talk))
• 17:35, 14 January 2026 Lptmswikids talk contribs deleted page Test-qin (content was: "=Detour: Extreme Value Statistics= Consider the REM spectrum of <math>M</math> energies <math>E_1, \dots, E_M</math> 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> We also define: <center> <math>E_{\min} = \...", and the only contributor was "Lptmswikids" (talk))
• 17:35, 14 January 2026 Lptmswikids talk contribs deleted page More general REM and systems in Finite dimensions (content was: "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 gro...", and the only contributor was "Lptmswikids" (talk))
• 17:35, 14 January 2026 Lptmswikids talk contribs deleted page Phase Transition in the Random Energy Model (content was: "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</m...", and the only contributor was "Lptmswikids" (talk))
• 17:34, 14 January 2026 Lptmswikids talk contribs deleted page Random energy model (content was: "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...", and the only contributor was "Lptmswikids" (talk))
• 17:34, 14 January 2026 Lptmswikids talk contribs deleted page Spin glass Transition (content was: "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 atoms interact via the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction. This interaction has a random sign due to the random spatial dist...", and the only contributor was "Lptmswikids" (talk))
• 17:33, 14 January 2026 Lptmswikids talk contribs deleted page Edwards Anderson order parameter (content was: "Since <math>\overline{J} = 0</math>, the model does not exhibit spatial magnetic order, such as ferromagnetic or antiferromagnetic order. Instead, the idea is to distinguish between two phases: * Paramagnetic phase: Configurations are explored with all possible spin orientations. * Spin glass phase: Spin orientations are random but frozen (i.e., immobile). The gla...", and the only contributor was "Lptmswikids" (talk))
• 17:32, 14 January 2026 Lptmswikids talk contribs deleted page Edwards Anderson model (content was: "The first significant theoretical attempt to describe spin glasses is the Edwards-Anderson model. For simplicity, we will consider the Ising version of this model. Ising spins take two values, <math>\sigma_i = \pm 1</math>, and are located on a lattice with <math>N</math> sites, indexed by <math>i = 1, 2, \ldots, N</math>. The energy of the system is expressed as a...", and the only contributor was "Lptmswikids" (talk))
• 17:32, 14 January 2026 Lptmswikids talk contribs deleted page Detour: Extreme Value Statistics (content was: "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 proba...", and the only contributor was "Lptmswikids" (talk))
• 17:32, 14 January 2026 Lptmswikids talk contribs deleted page Density above the minimum (content was: "'''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{Pr...", and the only contributor was "Lptmswikids" (talk))
• 17:32, 14 January 2026 Lptmswikids talk contribs deleted page Behavior in Different Phases (content was: "* '''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> <m...", and the only contributor was "Lptmswikids" (talk))
• 17:29, 14 January 2026 Lptmswikids talk contribs deleted page Back to the REM (content was: "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<...", and the only contributor was "Lptmswikids" (talk))
• 17:29, 14 January 2026 Lptmswikids talk contribs deleted page A Concrete Example: The Gaussian Case (content was: "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}}^{...", and the only contributor was "Lptmswikids" (talk))
• 11:51, 14 January 2026 Lptmswikids talk contribs created page L-1-new (Created page with "= 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 atoms interact via the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction. This interaction has a random sign due to the random spatial distribution of Mn atoms within the non-magnetic metal. A freezing temperature, <math...")
• 11:48, 14 January 2026 Lptmswikids talk contribs created page Edwards Anderson order parameter (Created page with "Since <math>\overline{J} = 0</math>, the model does not exhibit spatial magnetic order, such as ferromagnetic or antiferromagnetic order. Instead, the idea is to distinguish between two phases: * Paramagnetic phase: Configurations are explored with all possible spin orientations. * Spin glass phase: Spin orientations are random but frozen (i.e., immobile). The glass phase is characterized by long-range correlations in time, despite the absence of long-range correlations...")
• 11:48, 14 January 2026 Lptmswikids talk contribs created page Edwards Anderson model (Created page with "The first significant theoretical attempt to describe spin glasses is the Edwards-Anderson model. For simplicity, we will consider the Ising version of this model. Ising spins take two values, <math>\sigma_i = \pm 1</math>, and are located on a lattice with <math>N</math> sites, indexed by <math>i = 1, 2, \ldots, N</math>. The energy of the system is expressed as a sum over nearest neighbors <math>\langle i, j \rangle</math>: <center><math> E = - \sum_{\langle i, j \ran...")
• 11:47, 14 January 2026 Lptmswikids talk contribs created page Spin glass Transition (Created page with "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 atoms interact via the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction. This interaction has a random sign due to the random spatial distribution of Mn atoms within the non-magnetic metal. A freezing temperature, <math>T_f</math>, separates the...")
• 11:37, 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...")
• 11:37, 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...")
• 11:36, 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...")
• 11:36, 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}\;...")
• 11:36, 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]...")
• 11:35, 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...")
• 11:35, 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...")
• 11:34, 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 -...")
• 11:34, 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...")
• 11:29, 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...")
• 23:00, 13 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:''' *...")
• 17:09, 19 January 2024 Lptmswikids talk contribs changed group membership for Rosso from (none) to administrator
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